quelques probl`emes de la théorie des syst`emes paraboliques

Quelques problemes

de la theorie des systemes

paraboliques degeneres

non-lineaires

et des lois de conservation

Boris P. ANDREIANOV

These de doctorat en Mathematiques et Applications

soutenue

a l’Universite de Franche-Comte, Besancon, France

le 20 janvier 2000

devant le jury compose de

Assia Benabdallah, Universite de Franche-Comte

Philippe Benilan, Universite de Franche-Comte (le Directeur)

Valery Galkin, Institut de l’Energie Nucleaire, Obninsk, Russie (Rapporteur)

Raphaele Herbin, Universite de Provence, Marseille (Rapporteur)

Louis Jeanjean, Universite de Franche-Comte

Evgenii Radkevich, Universite Lomonossov de Moscou, Russie

Denis Serre, Ecole Normale Superieure de Lyon (Rapporteur)

Petra Wittbold, Universite Louis Pasteur de Strasbourg.

Remerciements

Je crois sincerement avoir eu une chance extraordinaire. Pendant les six dernieres

annees, Stanislav Nikolaevich Kruzhkov, puis Philippe Benilan, m’ont mene dans le monde

de la recherche. Parmi tout ce qu’ils m’ont donne, il m’est difficile de choisir le plus

precieux. Peut-etre, est-ce le gout pour la beaute des mathematiques; la volonte de ne pas

s’arreter a mi-chemin, d’aller au fond des choses. Et surtout, la confiance qu’ils m’ont

accordee, au-dela de ce que j’aurais pu esperer. A Stanislav Kruzhkov et Philippe Benilan,

j’adresse ma premiere pensee et ma plus profonde gratitude.

Je remercie beaucoup Evgenii Radkevich. Son soutien a aide a la poursuite de ce travail

en co-tutelle, les discussions avec lui ont ete stimulantes.

Athanasios Tzavaras s’est interesse a mes resultats; il m’a suggere un des problemes

abordes dans cette these. Je l’en remercie.

Denis Serre a accepte de donner son avis sur ce memoire; son appreciation m’est

tres importante. De plus, il m’a fait l’honneur de presider le jury de these. Je lui suis

doublement reconnaissant.

Je suis reconnaissant a Valerii Galkin, pour son rapport sur cette these, pour avoir

pris part a ce jury, et pour tout le soutien qu’il m’a accorde a Moscou.

Mes remerciements vont a Raphaele Herbin, pour l’avis qu’elle a donne sur ce travail,

pour ses explications, pour avoir accepte de participer au jury.

Je remercie Brian Gilding pour avoir accepte de faire un rapport sur ce memoire, et

pour l’attention qu’il a pretee a mes tous premiers resultats.

Assia Benabdallah m’a fait l’honneur de participer a ce jury. De longues discussions

avec elle, son encouragement constant m’ont ete precieux. Je lui adresse ma profonde

reconnaissance.

Je dois une partie importante de ma these a la collaboration avec Petra Wittbold. Elle

a bien voulu s’interesser aux diverses parties de ce memoire et prendre part au jury. J’en

suis honore. Je la remercie, et je remercie Michael Gutnic. Notre travail a trois m’a

beaucoup appris; il a ete un plaisir pour moi.

Louis Jeanjean a bien voulu s’interesser aux questions abordees dans cette these et

participer a ce jury. Ses remarques me seront tres utiles dans l’avenir; je le remercie de

son engagement.

La grande disponibilite de Catherine Pagani, Catherine Vuillemenot, Monique Digu,

Jacques Vernerey, Jean-Daniel Tissot, Odile Henri, Nathalie Pasquet m’a facilite la tache;

leur gentillesse m’a touche. Je leur dis “merci”.

Le Labo de Mathematiques de Besancon m’a accueilli pendant trois annees. Ou que

je sois, je serai toujours nostalgique de son ambiance. J’en garderai des souvenirs, j’en

garderai des amis.

Je dis “merci” et “spasibo” a mes amis : ceux de Moscou, de Besancon, de St-

Peterbourg... On a partage de bons moments; leur encouragement m’a beaucoup aide.

Ma mere et mon pere m’ont apporte un soutien inestimable. Je leur dedie cette these.

Resume

Dans la premiere partie, on traite par l’approche de viscosite auto-similaire le probleme

de Riemann pour la loi de conservation scalaire et les systemes de type “dynamique des

gaz isentropiques” en coordonnees de Lagrange et d’Euler. Dans chacun des cas, cette

etude aboutit aux resultats d’existence et d’unicite de la solution “wave-fan admissi-

ble” pour toute fonction de flux continue. En particulier, on couvre le cas d’apparition

du vide dans la dynamique des gaz et le cas des problemes mixtes avec transitions de

phase. D’autre part, pour une loi de conservation scalaire multi-dimensionnelle avec une

fonction de flux continue on demontre l’existence des solutions entropiques generalisees

maximum et minimum dans le cadre L1 ∩ L∞ . On etudie la nature de ces solutions a

l’aide de la theorie des semi-groupes non-lineaires; ensuite, on etend quelques resultats

d’unicite dus a Benilan et Kruzhkov.

Dans la deuxieme partie, on traite de systemes elliptiques-paraboliques dont les coeffi-

cients peuvent dependre de (t, x) . On demontre un theoreme de continuite des solutions

variationnelles par rapport aux donnees et obtient ainsi le resultat d’existence de Alt et

Luckhaus sous des hypotheses plus faibles, tout en mettant en evidence l’essentiel de

leurs arguments. On applique ensuite les techniques developpees pour demontrer la con-

vergence des schemas de volumes finis pour un systeme modele fortement nonlineaire,

qui apparaıt dans la physique des milieux poreux. On propose ainsi une approche pour

la convergence des methodes de volumes finis, ou la preuve se fait par reduction du cas

discret au cas continu.

Mots cles :

lois de conservation, probleme de Riemann, viscosite auto-similaire, dynamique des gaz

isentropiques, solutions entropiques generalisees, semi-groupes non-lineaires, systemes

elliptiques-paraboliques, conditions de type Leray-Lions, methodes de volumes finis.

Abstract

In the first part, one treats by the self-similar viscosity approach the Riemann problem

for a scalar conservation law and the systems of the “isentropic gas dynamics” type in

the Lagrange and Euler coordinates. In each of the cases, the study yields existence

and uniqueness of a wave-fan admissible solution for all continuous flux function. In

particular, the situation when vaccuum appears in gas dynamics is covered, as well as

the case of problems of mixed type with phase transitions. On the other hand, for a

scalar multidimensional conservation law with continuous flux function the existence of

maximum and minimum generalized entropy solutions in the L1 ∩ L∞ framework is

proved. Using the nonlinear semigroup theory, one studies the nature of these solutions;

then one extends some uniqueness results of Benilan and Kruzhkov.

In the second part, one treats elliptic-parabolic systems with coefficients that may

depend on (t, x) . A theorem on continuity of variational solutions with respect to data

is proved. This yields the existence result of Alt and Luckhaus under weaker hypotheses,

while clarifying the essence of their arguments. Necessary techniques are developped;

next, they are applied to proving convergence of finite volume schemes for a model

strongly nonlinear system, which appears in the study of porous media. An approach for

convergence of finite volume methods is proposed, where the proof goes on by reduction

of the discret case to the continuous one.

Key words :

conservation laws, Riemann problem, self-similar viscosity, isentropic gas dynamics, gen-

eralized entropy solutions, nonlinear semigroups, elliptic-parabolic systems, Leray-Lions

type conditions, finite volume methods.

Table des matieres :

Introduction

Chapitre I Enonces des “quelques problemes”

et un resume des resultats obtenus . . . . . . . . . . . . . . . . . . . . . . 13. . . 19

Chapitre II Les “quelques problemes”

dans le contexte mathematique et physique . . . . . . . . . . . . . 21. . . 31

Part 1. Conservation Laws with Continuous Flux Function

Chapter 1.I The Riemann Problem for Scalar Conservation Law

with Continuous Flux Function:

the Self-Similar Viscosity Approach . . . . . . . . . . . . . . . . . . . . . 35. . . 43

Chapter 1.II The Riemann Problem for p-System

with Continuous Flux Function . . . . . . . . . . . . . . . . . . . . . . . . . . 45. . . 62

Chapter 1.III On Viscous Limit Solutions to the Riemann Problem

for the Equations of Isentropic Gas Dynamics

in Eulerian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63. . . 82

Chapter 1.IV L1 -Theory of Scalar Conservation Law

with Continuous Flux Function . . . . . . . . . . . . . . . . . . . . . . . . . . 83. . . 98

Part 2. Weak Solutions for Elliptic-Parabolic Systems

Chapter 2.I Elliptic-Parabolic problems: Existence and Continuity

with Respect to the Data of Weak Solutions . . . . . . . . . 101. . . 124

Chapter 2.II Convergence of Finite Volume Approximations

for a Nonlinear Elliptic-Parabolic Problem:

a Variational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125. . . 159

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163. . . 172

Introduction

CHAPITRE I

Enonces des “quelques problemes”

et un resume des resultats obtenus†

Partie 1.Les lois de conservation

avec une fonction de flux continue

On etudie l’admissibilite et l’unicite de solutions pour la loi de conservation scalaire et les

systemes de type “elasticite non-lineaire” et “dynamique des gaz isentropiques”. En general,

la fonction de flux est seulement continue.

• Chapitre 1.I Le probleme de Riemann pour les lois de conservation scalaires avec une

fonction de flux continue : l’approche par viscosite auto-similaire.

· On traite le probleme suivant :

(CLε)

Ut + f(U)x = εtUxx,

U |t=0 =

u−, x < 0

u+, x > 0,

U : IR+ × IR 7→ IR, u± ∈ IR,

avec f(·) : IR 7→ IR continue.

· Les references principales sont :

Riemann [R1860], Rayleigh [Ray10], Hopf [H50], Gelfand [G59], Kalashnikov [Ka59], Dafer-

mos [D73a, D89], Kruzhkov [K69a, K69b, K70a], Kruzhkov, P.A.Andreyanov [KPA75],

Chapitre 1.IV.

†On presente dans ce Chapitre l’essentiel des resultats de la these. Les enonces et conditions exactes se

trouvent dans les chapitres correspondants dans les Parties 1 et 2.

14 Introduction

· Dans ce chapitre, on se donne pour objectif :

obtenir la solution admissible de (CL0) comme la limite des solutions de (CLε) quand

ε ↓ 0 , en donnant ainsi une preuve directe d’unicite pour (CL0) ; presenter le “wave fan

admissibility criterion” comme un critere bien adapte au cas de fonction de flux qui n’est

pas reguliere.

· Les principaux resultats sont :

- reduction de (CLε) avec ε > 0 a un probleme aux limites singulier, mais bien pose,

pour une equation differentielle ordinaire (Propositions 1,2);

- existence et unicite pour (CLε) avec ε > 0 (Theorem 1);

- convergence des solutions de (CLε) vers une limite, donnee par une formule explicite,

lorsque ε ↓ 0 , ce qui donne une demonstration directe d’unicite d’une solution “wave-fan

admissible” pour le probleme de Riemann (Theorem 2);

• Chapitre 1.II Le probleme de Riemann pour les p-systemes avec une fonction de flux

continue.


(pSε)

Ut − Vx = 0

Vt − f(U)x = εtVxx,

(U, V )|t=0 =

(u−, v−), x < 0

(u+, v+), x > 0,

(U, V ) : IR+ × IR 7→ IR2,

u±, v± ∈ IR.

Dans un premier temps, f : IR 7→ IR est supposee continue strictement croissante

(Sections 1-3), puis la condition de monotonie est supprimee.


Dafermos [D74, D89], Leibovich [Le74], Krejcı,Straskraba [KrSt93], Tzavaras [Tz96],

James [Ja80], Shearer [Sh82], Slemrod [Sl89], Chapitre 1.I.


obtenir la solution admissible de (pS0) comme la limite des solutions de (pSε) lorsque

ε ↓ 0 ; inclure le cas hyperbolique-elliptique en toute generalite; expliciter le contexte

physique de l’admissibilite pour ce dernier probleme.


- caracterisation de solutions de (pSε) , ε > 0 , et reduction a un probleme aux limites

pour une equation differentielle ordinaire (Lemma 1, Propositions 1,2,4 ; puis Lemma 4,

Propositions 5,6);

- unicite pour (pSε) , ε > 0 , et existence sous des hypotheses de croissance de f(·) en

±∞ (Theorem 1; puis Theorem 2);

0.I. Les enonces des problemes et un resume des resultats obtenus 15

- convergence des solutions de (pSε) vers une limite, donnee par une formule explicite,

lorsque ε ↓ 0 (Theorem 1; puis Theorem 2);

- observation que le “Riemann solver” obtenu permet de resoudre (pS0) d’une maniere

unique, mais ne correspond qu’a un nombre restreint de problemes physiques avec f(·)non-monotone (Remark 4).

• Chapitre 1.III Sur les limites visqueuses auto-similaires comme solutions du probleme

de Riemann pour les equations de la dynamique des gaz isentropiques en

coordonnees d’Euler.


(GDε)

ρt + (ρu)x = 0

(ρu)t + (ρu2 + p(ρ))x = εtuxx,

(ρ, u)|t=0 =

(ρ−, u−), x < 0

(ρ+, u+), x > 0,

(ρ, u) : IR+ × IR 7→ IR+ × IR,

ρ± > 0, u± ∈ IR;

La fonction p : IR+ 7→ IR est supposee continue strictement croissante; en general, le

comportement de p(·) en zero n’interdit pas l’apparition du vide dans les solutions.


Dafermos [D89], Kim [Kim99], Slemrod, Tzavaras [SlTz89], Rozhdestvenskii, Janenko

[RoJa], Cheng, Hsiao [ChHs], Wagner [Wa87], Chapitres 1.I et 1.II.


construire les solutions de (GDε) avec ε > 0 qui peuvent contenir un point du vide, afin

de pouvoir resoudre (GD0) pour toutes donnees de Riemann; demontrer l’existence et

l’unicite pour (GDε) , ε > 0 ; obtenir la solution admissible de (GD0) comme la limite

des solutions de (GDε) lorsque ε ↓ 0 , et observer la formation du vide.


- caracterisation des solutions de (GDε) pour ε > 0 , description des solutions avec

et sans le vide, et reduction de (GDε) , ε > 0 , a un probleme aux limites pour une

equation differentielle ordinaire (Lemma 1, Propositions 1,2,3, Lemmae 4,5,6);

- unicite pour (GDε) , ε > 0 et existence sous des hypotheses de croissance de p(·) en

+∞ (Theorem 1);

- convergence des solutions de (GDε) vers une limite, donnee par une formule explicite,

lorsque ε ↓ 0 (Theorem 2); observation de la structure de la solution admissible de

(GD0) , conditions necessaires et suffisantes d’existence d’une solution de (GDε) (for-

mule (38)) et d’apparition du vide (formule (39)).

16 Introduction

• Chapitre 1.IV La theorie L1 des lois de conservation scalaires avec une fonction de flux

continue†.

· On traite les problemes suivants :

(CP )

∂u

∂t+ divx φ(u) = g

u|t=0 = f,u : (0, T )× IRN 7→ IR,

(Eq) u+ divx φ(u) = f, u : IRN 7→ IR,

avec φ : IR 7→ IRN continue. Le cadre fonctionnel principal est c + L1 ∩ L∞ , c ∈ IR .

L’unicite d’une solution (c’est-a-dire, solution entropique generalisee) pour (CP ) et pour

(Eq) reste un probleme ouvert pour N ≥ 2 .


Benilan, Kruzhkov [BK96], Kruzhkov, Panov [KP90], Kruzhkov [K69a, K69b, K70a],

Benilan [B72], Crandall, Liggett [CL71], Benilan, Crandall, Pazy [BCP].


demontrer l’existence de la solution maximum et la solution minimum pour (CP ) et

(Eq) ; etablir le lien entre les solutions des deux problemes dans le cadre de la theorie

des semi-groupes non-lineaires; etablir des resultats partiels d’unicite, en particulier en

generalisant ceux de [BK96]; argumenter en faveur d’unicite en general.


- l’existence d’une solution pour (Eq) et (CP ) dans le cadre L∞ (Lemma 2, Theo-

rem 6);

- l’existence de la solution maximum et la solution minimum pour (Eq) et (CP ) (The-

orem 1);

- la generation du semi-groupe “solution maximum de (CP)” par l’operateur “solution

maximum de (Eq) ”, accretif a domaine dense, et pareil pour les solutions minimum

(Proposition 2, Theorem 2, Remark 1);

- pour une fonction de flux φ(·) et une constante c donnees, l’equivalence de l’unicite

en general pour (Eq) et pour (CP ) (Corollary 1);

- pour une fonction de flux φ(·) donnee, l’unicite est garantie pour tout c ∈ IR , excepte

peut-etre un ensemble au plus denombrable (Propositions 1,3);

- l’unicite pour (Eq) et (CP ) dans le cas ou (N − 1) composantes de φ(·) sont

monotones, par recurrence sur la dimension d’espace N (Theorem 3);

- l’unicite pour (Eq) et (CP ) dans le cas ou les modules de continuite en zero des

composantes de φ(·+ c)− φ(c) satisfont la condition anisotropique de decroissance de

[KP90, BK96] (Theorem 4);

†Il s’agit d’un travail fait en commun avec Philippe Benilan et Stanislav N. Kruzhkov.


- malgre cela, et malgre le resultat de non-unicite dans le cadre L∞ de [KP90] qui

souligne la pertinence de la condition anisotropique dans L∞ , conclusion que la non-

unicite dans le cadre L1∩L∞ , le cas echeant, n’est pas intrinsequement liee au caractere

non-holderien de la fonction de flux (note en bas de page sous Theorem 4).

Partie 2.Solutions faibles pour

les systemes elliptiques-paraboliques

On etudie l’existence des solutions faibles et la convergence des solutions approchees pour

une classe de systemes paraboliques degeneres. La methode suit et eclaircit l’approche vari-

ationnelle suggeree par Alt et Luckhaus ([AL83]).

• Chapitre 2.I Problemes elliptiques-paraboliques : l’existence et la continuite par rapport

aux donnees de solutions faibles†.

· On traite les problemes du type suivant :

(Pr)

b(·, v)t = div a(·, v, Dv) + f(·, v) sur (0, T )× Ω ⊂ IR+ × IRd

b(·, v(·))|t=0 = u0 sur Ω

+ conditions aux limites mixtes

de Dirichlet h(·) et de Neumann g(·, v) sur (0, T )× ∂Ω.

Ici v : (0, T ) × Ω 7→ IRN , la fonction b : (0, T ) × Ω × IRN 7→ IRN est telle que

b(t, x, ·) est le gradient d’une fonction convexe differentiable Φ(t, x, ·) : IRN 7→ IR . On

prend des hypotheses appropriees sur les donnees b, a, f, g, h, u0 qui generalisent, pour

une partie, les conditions de [AL83] et [LJLL65] (les hypotheses (H1)−(H9) en general,

les hypotheses (1)−(3) dans le cas simple avec f, g, h qui s’annulent completement et

sans dependance en (t, x) ).


Alt, Luckhaus [ALpr, AL83], J.-L.Lions [JLL], Leray, J.-L.Lions [LJLL65], Kruzhkov [K69a],

Benilan, Wittbold [BW99].


obtenir un resultat d’existence et de dependance en donnees de solutions faibles pour (Pr)

avec les coefficients qui peuvent dependre de (t, x) ; mettre en evidence les arguments

essentiels dans la methode variationnelle proposee dans [ALpr, AL83]; discuter la necessite

des conditions de structure sur a , f , Φt .

†Il s’agit d’un travail fait en commun avec Philippe Benilan.

18 Introduction


- une version du “chain rule” lemme, c’est-a-dire

(ChR)d

dt

∫

Ω

B(t, ·, v(t, ·)) =< b(t, ·, v(t, ·))t, v(t, ·) >E′,E −∫

Ω

Φt(t, ·, v(t, ·)),ou E est l’espace ou l’on cherche les solutions variationnelles, E ′ est son dual et

B(t, x, z) est

∫ z

0

(b(z)− b(ζ)) dζ (Lemma 1);

- une version de l’argument de compacite de [K69a], qui suggere que, pour une famille

de solutions d’equations d’evolution, la compacite dans L1 en (t, x) se deduit d’une

estimation L1 sur les translatees en x des solutions, a condition d’avoir des estimations

uniformes sur les solutions dans L1 d’une part, et sur la partie droite des equations dans

un espace de Sobolev negatif d’autre part (Lemma 6);

- une version appropriee de l’argument de Minty-Browder (Lemma 7);

- un theoreme de continuite pour les solutions faibles par rapport aux donnees de (Pr) ,

dans une topologie naturelle vis-a-vis des hypotheses prises, sous des hypotheses de

structure supplementaires (H11) − (H13) sur a , f et Φt et dans le cas h = 0 ;

dans la preuve de ce resultat, on applique successivement les trois arguments ci-dessus

et un argument qui combine l’equi-integrabilite avec le theoreme de Egorov (Theorem 1,

Remark 3);

- corollaires d’existence d’une solution faible pour (Pr) (Corollaires 1,2,3, Remark 4);

- indications sur la non-pertinence des restrictions imposes par les conditions de structure

(H11), (H12) pour l’existence d’une solution faible.

• Chapitre 2.II La convergence des approximations par les methodes des volumes finis

pour un probleme elliptique-parabolique non-lineaire : une approche

variationnelle†.


(pL)

b(v)t = div ap(Dv) sur (0, T )× Ω ⊂ IR+ × IRd

b(v)|t=0 = u0 sur Ω

v = 0 sur ∂Ω,

pour un domaine polyhedral Ω ⊂ IRd , ou b est le gradient d’une fonction convexe

differentiable Φ : IRN 7→ IRN et div ap(D·) est un p-laplacien.


Alt, Luckhaus [ALpr, AL83], J.-L.Lions [JLL], Eymard, Gallouet, Herbin [EyGaHe], Chapitre

2.I

†Il s’agit d’un travail fait en commun avec Michael Gutnic et Petra Wittbold.



ecrire un schema de volumes finis pour (pL) , demontrer la consistance du schema et

proposer une approche de preuve de la convergence qui cherche a reduire le probleme au

cas continu du Chapitre 2.I autant que possible.


- proposition d’une classe de schemas de volumes finis qui approche le gradient d’une

maniere satisfaisante vis-a-vis de la non-linearite de ap , puis d’un exemple d’un tel

schema (Definition 4, Remark 2);

- existence d’une solution discrete et estimations a priori (Theorem 2, Proposition 1);

- introduction des approximations “continues” de la solution discrete vh , du terme parabo-

lique b(vh) et du terme elliptique ap(“Dvh”) ayant de “bonnes proprietes”; reecriture

du systeme d’equations algebriques issu du schema des volumes finis sous une forme

d’equation dans D′ (Lemmae 2,5,6, Proposition 2);

- consistance de l’approximation par la methode des volumes finis de l’operateur div ap(D·)pour la classe de schemas proposee (Definition 5, Proposition 3, Theorem 3);

- convergence de vh , lorsque le pas de discretisation tend vers zero, vers une solution

faible de (pL) dans la classe de schemas proposee, qui est demontree par la methode

variationnelle du Chapitre 2.I appliquee aux approximations “continues” construites au-

paravant (Theorem 1).

CHAPITRE II

Les “quelques problemes” dans le contexte

mathematique et physique†

1. La theorie des equations aux derivees partielles, s’il y en a une, a pour points de depart “les

trois baleines”: les trois equations classiques. Ce sont l’equation hyperbolique (H) utt =

∆u , dite equation d’ondes; l’equation parabolique (P ) ut = ∆u , dite equation de la

chaleur; l’equation elliptique (E) ∆u = 0 , dite equation de Laplace. Il est habituel de

parler de caractere hyperbolique, parabolique ou elliptique dans des problemes plus generaux,

en se basant sur la ressemblance des methodes appliquees et le comportement des solutions.

Pour chacun des archetypes (H) , (P ) , (E) il existe un cadre specifique et une theorie

mathematique a part entiere, qui observe des phenomenes qui lui sont propres. Ainsi, (H)

et (P ) decrivent l’evolution en temps, tandis que (E) est relatif a la stabilite en espace.

Il y a un effet regularisant pour (P ) et (E) , c’est-a-dire, la regularite des solutions est

meilleure que celle des donnees; cela n’est pas le cas de (H) . Tandis que l’on observe

dans (P ) les phenomenes de propagation a vitesse infinie et de dissipation d’energie, (H)

presente les proprietes de conservation d’energie et de domaine de dependance fini. On

pourrait enumerer d’autres differences substantielles entre (H) , (P ) et (E) , et pourtant

ces equations lineaires sont les modeles les plus simples de phenomenes reels qui surgissent

en physique. Quand on veut concevoir des modeles plus generaux ou plus realistes, on

est souvent amene a considerer des problemes qui sont, d’une part non-lineaires, et d’autre

part, qui melangent en eux les comportements typiques pour (H) , (P ) , (E) . Le caractere

non-lineaire restreint dramatiquement les outils qui ont ete concus lors du traitement de

(H) , (P ) , (E) , et amene a en utiliser de nouveaux dans chaque cas qui se presente.

†On se donne pour objectif de presenter ici une vue generale sur l’ensemble des sujets abordes dans cette

these. Deux points de vue s’imposent : celui du contexte mathematique des equations aux derivees partielles

non-lineaires, mais aussi celui des motivations et des modeles physiques qui pesent considerablement sur le

sujet.

Apres un petit avant-propos, on fait le tour d’horizon des problemes traites. Il est suivi par une discussion

des motivations, methodes et resultats de chacune des deux parties de cette these. En conclusion, on explicite

un certain nombre des points d’interrogation que souleve le travail presente a Votre attention.

22 Introduction

Ainsi, j’ai ete confronte dans l’etude des problemes cernes dans le Chapitre 1 ci-dessus a

un nombre de difficultes qui proviennent de la non-linearite, de la non-regularite des donnees,

de changement ou d’imprecision sur le type des equations etudiees.

2. Les systemes hyperboliques des lois de conservation font l’objet de la premiere partie de

la these. La loi scalaire (CP ) , etudiee dans les chapitres 1.I et 1.IV, est tres interessante

en elle-meme. Elle correspond a un certain nombre de phenomenes, elle peut modeliser le

traffic routier par exemple; et elle represente le premier pas vers l’etude de systemes aussi

importants que ceux de la dynamique des gaz. Dans les chapitres 1.II et 1.III, on traite

le probleme qui a ete pose et resolu pour la premiere fois par B.Riemann ([R1860]) et qui

consiste a poursuivre la resolution du systeme de la dynamique des gaz isentropiques en

presence de discontinuites. Les systemes (pS0) , (GD0) concernes par les chapitres 1.II et

1.III modelisent les ecoulements isentropiques des gaz parfaits en coordonnees de Lagrange

et d’Euler, mais aussi l’elasticite non-lineaire et les fluides de van der Waals (cf. [R1860,

Ray10, CouFr, RoJa, Se, ChHs, Tz98, Ja80, Pe87]). Dans ce dernier cas en particulier, qu’on

traite en coordonnees de Lagrange, on est contraint a renoncer au caractere hyperbolique

du systeme, a cause de la presence des phases hyperboliques et elliptiques et des transitions

entre elles. Dans les coordonnees d’Euler, on considere aussi les cas d’apparition de l’etat du

du vide dans les solutions; le vide est la degenerescence specifique a certains systemes de la

dynamique des gaz. L’approche employee dans les chapitres 1.I-1.III necessite l’obtention de

resultats d’existence et d’unicite pour les problemes de Riemann (CLε) , (pSε) , (GDε) qui

sont les systemes regularises avec un terme parabolique lui-meme degenere.

De plus, on traite dans (CL0) , (pS0) , (GD0) et (CP ) le cas de fonction de flux

seulement continue. Ceci n’est en rien pathologique; en effet, une equation aussi simple

que (u3)t + (u2)x = 0 donne lieu a la loi de conservation scalaire wt + f(w)x = 0 avec

f : w 7→ w2/3 , qui n’est pas une fonction localement lipschitzienne. Le manque de regularite

de la fonction de flux entraıne des effets extremement interessants dans le cas d’une loi de

conservation scalaire multidimensionnelle. On observe le phenomene de propagation a vitesse

infinie, ce qui peut entraıner la perte du caractere conservatif de l’equation et la non-unicite

d’une solution dans la classe (pourtant tres bonne, cf. [K69a, K69b, K70a, PA71, KH74,

KPA75]) de solutions entropiques generalisees dans L∞ (cf. [B72, KP90, BK96]). La non-

unicite n’est peut-etre pas le cas dans le cadre de solutions L1 ∩ L∞ ; l’investigation de ce

dernier probleme, couplee avec une etude de l’equation (Eq) , et qui developpe les resultats

de [B72, KP90, BK96], fait l’objet du chapitre 1.IV.

D’un autre cote, l’unicite pour le probleme de Riemann dans le cas scalaire uni-dimension-

nelle (CL0) , pour une classe de solutions verifiant un critere d’admissibilite usuel depuis

[D74, D89] (et qui remonte aux idees de [Ray10, H50, G59, Ka59]), est etudiee directement

dans le chapitre 1.I. Des etudes tres similaires sont menes a bien dans chacun des chapitres 1.II

et 1.III. Ceci etend en particulier les resultats de [Ka59, Tz95, Sh82, Kim99].

0.II. Sur le contexte mathematique et physique 23

Des problemes lies aux degenerescences et singularites se presentent egalement dans les

equations des milieux poreux (cf. [Bear, AL83]). Les systemes que l’on considere peuvent

en particulier contenir des phases purement elliptiques. Par exemple, pour l’equation de

Richards, qui trouve d’importantes applications dans les sciences d’environnement, le terme

parabolique atteint le regime stationnaire en 1 (l’etat de saturation). Dans les regions

de degenerescence elliptique, l’evolution en temps d’une solution n’est que partiellement

controlee, ce qui ne permet pas d’obtenir de la compacite forte des solutions la ou c’etait le

cas pour les systemes paraboliques non-degeneres. Cela complique en particulier la question

de l’existence des solutions.

On fait une etude d’existence et de dependance continue pour les solutions faibles des

systemes elliptiques-paraboliques (Pr) dans le chapitre 2.I, en etendant en particulier le

resultat d’existence de [AL83]. On espere avoir atteint une plus grande clarte dans l’exposition

des arguments qui menent a l’existence d’une solution faible pour (Pr) , par rapport au papier

[AL83] qui est ici la reference cruciale.

Vu l’interet que suscite le developpement de schemas numeriques pour la modelisation

d’ecoulements dans les milieux poreux, on aborde dans le chapitre 2.II la convergence pour

les methodes des volumes finis, employees frequemment dans ce domaine (cf. [EyGaHe]).

On teste la convergence sur l’exemple d’une equation (en fait, d’un systeme) (pL) qui est

une forme simplifiee du modele de filtration d’un fluide en regime turbulent a travers un

milieu poreux, gouvernee par une loi de Darcy non-lineaire, ou encore d’ecoulement d’un gaz

turbulent dans un pipeline (cf. [DiDT94] et leurs references). A ma connaissance, il n’existe

pas d’autre resultat de convergence de schemas de volumes finis pour des equations de type

(pL) et, plus generalement, (Pr) dans le cas p 6= 2 ; en etudiant (pL) , on cherche a

debroussailler le chemin pour pouvoir aborder un jour des cas plus realistes.

3. Un probleme essentiel de la theorie des systemes hyperboliques des lois de conservation est

celui de l’unicite. C’est la theorie des solutions entropiques generalisees, dites g.e.s., qui resout

ce probleme pour les lois de conservation scalaires multi-dimensionnelles. La definition et les

methodes classiques, elaborees dans [K69a, K69b, K70a], donnent une theorie mathematique

tout a fait satisfaisante, au moins dans le cas de fonction de flux localement lipschitzienne.

On reviendra sur ce sujet dans la partie consacree au chapitre 1.IV. Avant, on aborde une

autre facette de la question de l’unicite pour les lois et systemes de conservation, qui est les

conditions d’admissibilite des chocs. Notons que la definition d’une g.e.s. se traduit par une

telle condition dans les solutions continues par morceaux (ou, plus generalement, dans les

solutions a variation localement bornee). Mais les conditions d’admissibilite des chocs sont

anterieures a la notion d’une g.e.s. (cf. [H50, O57, Lax57, G59]), et les deux sont reliees

par une motivation commune qui est la methode de viscosite. Ceci est important, vu les

recents developpements dans le domaine des chocs non-classiques (cf. [LF98]). On peut, en

effet, construire des theories non-classiques pour certaines lois de conservation non-convexes,

24 Introduction

et ou les solutions s’obtiennent comme limites d’approximations par la diffusion couplee a

une dispersion lorsque les deux tendent vers zero. Tout ce qui a ete dit exige une certaine

explication, ou plutot un detour historique. Je ne donne pas les references a beaucoup

de travaux importants qui ont amene a une bonne position du probleme mathematique de

resolution des lois de conservation; les noms des principaux intervenants et leurs contributions

sont contes dans l’introduction de l’article [D89].

La difficulte de la resolution globale en temps des systemes de la dynamique des gaz est

connue depuis le XIX siecle. Elle surgit a cause du caractere non-lineaire de propagation

d’ondes. A la difference de l’equation d’ondes (H), ou la regularite des donnees initiales

est preservee (faute d’effet regularisant), les solutions d’une loi de conservation non-lineaire

peuvent developper des discontinuites en temps fini meme lorsque les donnees initiales sont

analytiques. B.Riemann ([R1860]) semble etre le premier qui accepte des solutions discon-

tinues dans la dynamique des gaz isentropiques et construit une solution pour le probleme

d’evolution d’une discontinuite elementaire, qui porte son nom. En particulier, il ne choisit

pour la construction de la solution que “la moitie” des chocs envisageables, en pressentant

ainsi la condition d’admissibilite de Lax (cf. [Lax57]). Un bref resume du travail de Riemann

est donne dans [Se, Chap.4.6]. L’idee n’a pas fait l’unanimite. Rayleigh ([Ray10]) ecrit a pro-

pos de la condition de Rankine-Hugoniot, qui decrit les chocs envisageables dans une solution

discontinue: “however valid <the Rankine-Hugoniot condition> may be, its fulfillment

does not secure that the wave so defined is possible. As a matter of fact, a whole class

of such waves is certainly impossible, and I would maintain, further, that a wave of the

kind is never possible under the conditions, laid down by Hugoniot, of no viscosity or

heat conduction.†” C’est en retablissant une petite viscosite et/ou conductivite que Rayleigh

donne un sens a la distinction entre les “regimes permanents” (les chocs) admissibles et

inadmissibles; distinction qui n’a pas ete remarquee par Rankine lorsque ce dernier avait

deduit ses conditions de choc en partant des fluides conductifs. Le point de vue que defend

Rayleigh est, bien sur, celui du mecanicien. Il ne distingue pas le probleme mathematique du

phenomene physique que ce probleme est appele a decrire. La notion meme d’une solution

discontinue lui est etrangere: les “regimes permanents” ne sont pour lui rien d’autre que des

ondes de compression dans une region d’espace tres petite, et qui sont entretenues par la

dissipation d’energie dans la zone de transition.

Bien sur, aujourd’hui on peut donner tres facilement un sens purement mathematique a

un systeme de lois de conservation. C’est le sens faible, d’ailleurs plus naturel qu’une relation

differentielle dans le contexte physique de la conservation locale des grandeurs extensives.

On peut trouver beaucoup de solutions discontinues dans ce sens pour un meme probleme

†“aussi valide qu’elle < la condition de Rankine-Hugoniot> soit, rien ne garantit qu’une onde ainsi definie

soit possible dans le cas ou elle < la condition de Rankine-Hugoniot> est remplie. En fait, la classe entiere

de ces solutions est certainement impossible, et je maintiendrai, en outre, qu’une onde de la sorte n’est jamais

possible sous les conditions, imposees par Hugoniot, d’absence de viscosite et de transmission de la chaleur.”

de Riemann, de sorte qu’il n’y a pas d’unicite. Ce qui distingue les differentes solutions

dans le cas de la dynamique des gaz, c’est leurs proprietes vis-a-vis de la conservation de

l’energie. Rayleigh semble etre le premier a observer que “maintenance of type in such a

<shock> wave <of condensation> requires removal of energy from the wave, while in

the contrary case of <shock> wave of rarefaction additional energy would need to be

supplied‡.” Ce qui permet de trancher, dans le cadre de modele considere, en accord avec

le deuxieme principe de thermodynamique dont l’importance avait ete negligee auparavant:

“although dissipative forces, such as those arising from viscosity, may possibly constitute

a machinery capable of maintaining the type of <shock> wave of condensation, in no

case they can maintain the type of <shock> wave of rarefaction§.” Du point de vue

moderne, on peut dire que Rayleigh postule implicitement que seules les limites des solutions

du systeme prenant en compte de petits effets dissipatifs sont admissibles comme solutions

faibles du systeme de la dynamique des gaz isentropiques. C’est de la que la theorie classique

des lois de conservation est partie quarante ans plus tard, avec le papier de Hopf [H50]

sur l’equation ut + (u2)x = 0 . Une solution faible du probleme de Cauchy pour cette

equation est construite comme la limite des solutions du meme probleme pour l’equation

ut + (u2)x = εuxx , que l’on peut resoudre explicitement. Il resulte du travail de Hopf que

seules les chocs qui joignent un etat u− a gauche a un etat inferieur u+ a droite sont

admissibles. Un moyen facile de “predire” ce dernier resultat est de considerer la possibilite

d’approximation de chocs par les ondes planes, approche qui fait d’ailleurs le titre de ce

meme papier de Rayleigh [Ray10]. Dans [G59], Gelfand montre deja comment resoudre le

probleme de Riemann pour une loi de conservation d’une maniere unique, en partant de cette

caracterisation par les ondes planes. Une autre variante de l’idee de viscosite apparait chez

Kalashnikov ([Ka59]). C’est cette version, vue a travers l’interpretation de Dafermos ([D89]),

qui donne lieu au chapitre 1.I, puis 1.II et 1.III ci-dessous.

C’est toujours l’idee de viscosite qui tend vers zero qui suggere la pertinence de la definition

d’une g.e.s. de Kruzhkov. Cette definition propose une caracterisation intrinseque d’une

solution; la motivation reste exterieure (Kruzhkov l’ecrit tres clairement dans ses notes de

cours [K70b], dont la diffusion reste malheureusement tres limitee). La theorie des g.e.s.

donne une reponse parfaite aux questions d’existence, d’unicite et de dependance continue

pour le cas d’une loi de conservation a fonction de flux localement lipschitzienne, dans la

classe de fonctions bornees, et cela en plusieurs dimensions d’espace. Helas, la generalisation

aux systemes de l’approche entropique sous forme de Lax (cf. [K70a, Lax71]) rencontre

‡“la maintenance d’une telle onde <de choc de condensation> necessite d’enlever de l’energie a l’onde,

tandis que dans le cas contraire d’une onde <de choc> de detente il faudrait lui livrer de l’energie addition-

nelle.”§“bien que les forces dissipatives, comme celles qui proviennent de la viscosite, pourraient constituer une

machinerie capable de maintenir le type d’onde <de choc> de condensation, en aucun cas elles ne peuvent

maintenir le type d’onde <de choc> de detente.”

26 Introduction

des difficultes importantes dans le cas des systemes 2× 2 , et incontournables au-dela. Ceci

incite a reprendre l’etude d’admissibilite des chocs. Or, on sait actuellement (cf. [Br96, Br99])

que la bonne condition d’admissibilite de chocs (cf. [Lax57]) pour la classe tres etudiee de

systemes vraiment nonlineaires ou lineairement degeneres (cf. [Lax57, Sm, Se]) peut en elle

seule donner l’unicite d’une solution. La question de l’unicite, tout comme la question de

l’existence (cf. [Gl65, Br99]), peut en fait etre ramenee a l’admissibilite pour les problemes

de Riemann. Pour les systemes strictement hyperboliques, il existe une construction (cf.

[Liu75, Liu81]) qui permet la resolution d’une maniere unique d’un probleme de Riemann

lorsque le saut dans les donnees est petit. La resolution globale peut etre beaucoup plus

compliquee dans chaque cas qui se presente.

Or, c’est une approche tres bien adaptee pour la resolution du probleme de Riemann que

propose Dafermos dans [D73a, D74, D89], et qui d’ailleurs permet de juger admissible ou non

une solution toute entiere, plutot que de la considerer choc par choc. L’unique solution d’un

probleme de Riemann ne peut dependre que du quotient x/t ; autrement dit, la solution est

auto-similaire. L’addition d’une viscosite comme celle dans (CLε) , (pSε) , (GDε) permet de

preserver cette propriete. L’idee de l’approche par viscosite auto-similaire est donc de ramener

la discussion d’un probleme de Riemann au niveau des equations differentielles ordinaires; aussi

une variete de methodes s’applique (cf. [D74, DDp76, Tz96], et [Tz98] pour un etat des lieux

dans ce domaine). Le programme habituel est de demontrer, par des methodes topologiques,

l’existence pour le probleme regularise; obtenir des estimations sur la variation des solutions,

puis passer a la limite par compacite; enfin, etudier la structure des limites vis-a-vis de la

possibilite d’approximation des ondes de choc par les ondes planes.

Tout en profitant de la forme auto-similaire, mais egalement de la structure elementaire

des systemes particuliers etudies, je propose dans les chapitres 1.I-1.III une approche differente

d’investigation des solutions. Les moyens mathematiques employes ici sont simples, pour

ne pas dire rudimentaires. Par une analyse de plus en plus laborieuse, on est amene dans

chacun des cas (CLε) , (pSε) , (GDε) a resoudre un probleme aux limites pour une equation

differentielle ordinaire. A chaque fois, cette equation admet un principe de maximum; celui-

ci me permet d’etablir des resultats tres precis d’existence, d’unicite et de convergence des

solutions lorsque ε ↓ 0 . Ainsi on demontre dans chaque cas que la viscosite auto-similaire

choisit une unique solution pour les problemes de Riemann. En plus, les formules explicites

(certes, connues) pour les solutions limites, que l’on obtient directement, permettent de

distinguer les chocs qui ne peuvent apparaıtre dans aucune solution et qu’on peut donc juger

comme inadmissibles; ceci remplace l’etude par les ondes planes.

Vu ce qui a ete dit sur le bien-fonde de la definition d’une g.e.s. et l’origine commune

des differentes branches de la theorie des systemes des lois de conservation, il est etonnant

qu’il existe une methode purement mathematique qui donne un sens intrinseque aux lois

de conservation. C’est d’ailleurs un des deux traits d’union entre les deux parties de cette


these. Cette methode qui s’applique en meme temps aux lois de conservation scalaires et

aux equations elliptiques-paraboliques est l’approche par les semi-groupes non-lineaires. D’un

point de vue pratique, elle consiste a construire les solutions des equations d’evolution par le

procede de discretisation implicite en temps. Le trait commun des operateurs associes a une

loi de conservation scalaire et a une equation elliptique-parabolique coercive est la propriete

d’accretivite, qui donne la possibilite de continuer l’approximation a chaque pas de temps

d’une maniere stable. La theorie de semi-groupes non-lineaires peut permettre d’etablir

l’existence; mais en outre, elle peut donner l’unicite d’une solution dans les cas ou cette

unicite est un veritable probleme, (CP ) par exemple. Le procede meme de construction

des solutions semi-groupes (cf. [BCP]) inscrit la condition d’admissibilite dans les solutions;

l’irreversibilite en temps, suggeree dans les modeles physiques par la deuxieme loi de la

thermodynamique, est ainsi capturee (cf. [BCP88]).

Les resultats du chapitre 1.IV marquent davantage le lien qui existe entre les solutions

entropiques generalisees (cf. [K69a, K69b, K70a]) et les solutions semi-groupes (cf. [B72])

des lois de conservation. On utilise l’approche par semi-groupes non-lineaires pour etablir

la generation des semi-groupes des g.e.s. maximum, minimum de (CP ) par les operateurs

de la g.e.s. maximum, minimum de (Eq) , respectivement. On en deduit l’equivalence

entre l’unicite pour (CP ) et l’unicite pour (Eq) . Cela s’applique a la demonstration du

Theoreme 3 qui se fait par recurrence sur la dimension N d’espace et qui est le resultat

le plus interessant du chapitre 1.IV. Les autres techniques employees sont essentiellement

celles du papier [BK96]. L’existence des g.e.s. maximum et minimum, en absence presumee

d’unicite lorsque la fonction de flux n’est pas localement lipschitzienne, n’est pas un fait

trivial; on la prouve ici pour des donnees qui tendent vers une constante a l’infini. Comme

pour beaucoup d’autres resultats de ce travail, la demonstration est basee sur la possibilite

de prendre la fonction caracteristique de IRN tout entier comme la fonction test dans une

inegalite de type Kato (Lemma 3.1 dans [BK96]). Recemment, le resultat d’existence des

solutions maximum et minimum pour (CP ) et pour (Eq) a ete etendu au cas des donnees

bornees generales (cf. [P..]). Cela est d’autant plus interessant que l’on connaıt un exemple

de non-unicite (cf. [KP90]). Par contre, on remarque en passant que la non-unicite, le cas

echeant, d’une g.e.s. pour (CP ) et (Eq) dans le cadre L1 ∩L∞ doit etre gouvernee par

une propriete fine de la fonction de flux.

4. Jusqu’il y a quelques annees, l’unicite pour les systemes hyperboliques de lois de con-

servation manquait de resultats aussi importants que celui de Kruzhkov dans le cas scalaire.

C’est une analyse tres engagee de la structure d’ondes qui a permis a Bressan et ses col-

laborateurs ([Br96, Br99]) d’apporter une lumiere de comprehension a cette question. En

effet, j’ai deja mentionne les difficultes de l’approche entropique; l’approche par les semi-

groupes non-lineaires est freinee par le fait que l’on ne connaıt pas de norme pour laquelle

l’operateur associe au probleme d’evolution soit accretif. Cette meme difficulte surgit dans

28 Introduction

l’etude d’unicite pour les systemes elliptiques-paraboliques de type (Pr) , tandis que dans le

cas scalaire une reponse satisfaisante a la question d’unicite est donnee dans [Ot96, CaW99]

et [Ca99]. Dans ce cadre, la non-unicite, le cas echeant, est due a la degenerescence de la

partie parabolique et non a l’irregularite generique des solutions, comme c’est le cas pour les

systemes hyperboliques de lois de conservation.

Mais c’est encore l’existence d’une solution faible pour (Pr) qui souleve des questions.

La methode classique de construction d’une solution variationnelle (cf. [AL83]) exige les

hypotheses supplementaires (dites conditions de structure) de la dependance des coefficients

a, f en b(z) et non en z tout simplement. Or, l’approche par les semi-groupes non-lineaires

en particulier montre que ces conditions ne sont pas toujours indispensables (cf. [BW99]).

De plus, on trouve des indications a la meme conclusion tout en restant dans le cadre de

l’approche variationnelle de Alt-Luckhaus ([AL83]), qui est la methode du chapitre 2.I. Pas

tres importante du point de vue des applications physiques, la question de pertinence des

conditions de structure n’en reste pas moins obsedante. Je n’ai pas reussi a la resoudre,

bien qu’un travail important a ete fait dans le chapitre 2.I pour “minimiser” les restrictions

imposes sur (Pr) par l’approche de Alt-Luckhaus.

Cette methode prend sa premiere origine dans le travail [Bro63], qui se base sur la possi-

bilite d’appliquer l’approche de monotonie introduite dans [Mi62, Mi63] a la construction de

solutions aux equations elliptiques. Cette idee a ete etendue a une methode “de monotonie

et compacite” (cf. [JLL]) qui permet de resoudre par l’approche variationnelle, a travers la

methode de Galerkin, une classe de problemes elliptiques ([LJLL65]) et paraboliques ([JLL]).

Les conditions de Leray-Lions reapparaissent dans tout le procede de developpement de cette

approche variationnelle.

Dans le cas tres important de degenerescence elliptique d’un systeme parabolique, deux

autres arguments essentiels ont ete apportes dans [AL83]. Le premier (cf. aussi [Bam77])

impose a la fonction b dans (Pr) d’etre le gradient en z d’une fonction convexe; c’est le

“chain rule” lemme de type (ChR) . Il permet d’obtenir dans un premier temps les estima-

tions a priori qui remplacent celles de Leray-Lions, et de passer a la limite par l’argument de

Minty-Browder dans la phase finale de la demonstration. L’autre restriction que l’utilisation

de ce lemme impose se traduit dans les intrepretations physiques par la finitude de l’energie

dans l’etat initial. Le deuxieme argument est celui de la compacite L1 en temps et en espace

non pas pour les solutions v , mais pour les termes paraboliques b(v) . C’est ce deuxieme

argument qui est remplace dans le chapitre 2.I par un lemme assez ancien ([K69a]) et tres

bien adapte a la question qu’on se pose. L’utilisation de ce dernier lemme, qui est un resultat

de compacite tres general pour les equations d’evolution, sert aussi de trait d’union entre les

deux parties de cette these. C’est en effet ce lemme qui rend facile l’existence d’une g.e.s.

bornee pour le probleme de Cauchy pour une loi de conservation scalaire avec une fonction de

flux continue; une demonstration ecrite de ce resultat, qui semble manquer, est donnee dans


les annexes du chapitre 1.IV. De la meme maniere, on donne dans les annexes du chapitre 2.I

les versions appropriees des trois arguments qui menent a l’existence pour (Pr) : le “chain

rule” lemme, le lemme de compacite dans L1 et l’argument de Minty-Browder.

Parallelement a la simplification de la preuve, on couvre dans le chapitre 2.I le cas de

dependance des coefficients dans (Pr) en (t, x) , en particulier de la dependance de b

en t (qui s’avere delicate, au point de donner lieu a un probleme qui est reste ouvert).

A ce moment un quatrieme argument apparaıt, une combinaison d’equi-integrabilite des

termes dans l’equation avec le theoreme de Egorov, et qui ramene la discussion au niveau

de coefficients uniformement continus en l’ensemble des variables. Quant aux conditions

de type Leray-Lions, on essaie d’imposer les hypotheses les moins restrictives dans le cadre

Lp(0, T ;W 1,p(Ω; IRN)) , choisi pour les solutions faibles. Une etude de (Pr) dans les espaces

d’Orlitz, a l’instar de [Kac90], doit permettre d’affaiblir certaines restrictions. Notons qu’on

ne traite pas dans le chapitre 2.I en toute generalite le cas ou la condition au bord sur

une des composantes de la solution est une condition de Neumann pure. D’une part, cela

exigerait une hypothese supplementaire sur la fonction b . D’autre part, bien que l’essentiel

de nos arguments s’applique aussi dans cette situation, on ne peut pas inclure ce cas dans le

theoreme principal sans nuire davantage a la lisibilite de la preuve.

Le resultat principal qu’on demontre dans le chapitre 2.I est que l’ensemble des solu-

tions faibles de (Pr) (faute d’en connaıtre l’unicite) est semi-continu inferieurement par

rapport aux perturbations des coefficients et des donnees de (Pr) dans la topologie na-

turelle. L’existence est obtenue comme un corollaire de ce resultat, en utilisant la methode

de Galerkin tout comme dans [JLL] and [ALpr]. C’est ce meme theoreme de continuite qui

donne la convergence des approximations de Galerkin.

Des problemes elliptiques-paraboliques modelisent, en particulier, des phenomenes qui

interessent l’industrie petroliere. D’ou l’importance des aspects numeriques. Les methodes

classiques utilises en resolution numerique de tels systemes, du type elements finis, ont un

inconvenient certain. C’est l’absence du caractere conservatif local. Par contre, les methodes

de type volumes finis sont concus pour assurer cette conservativite (cf. [EyGaHe]). A la

difference des methodes de Galerkin, on ne peut pas directement appliquer les resultats de

[AL83] ou du chapitre 2.I pour les volumes finis. Dans le chapitre 2.II, on s’est donne

pour objectif d’adapter le resultat du chapitre 2.I au cas d’approximation du systeme par les

methodes des volumes finis.

On a choisi le systeme (pL) , fortement non-lineaire, pour montrer la possibilite d’une

telle adaptation. Le chemin qui est souvent emprunte pour aborder un tel probleme est la

“discretisation” des arguments de la preuve du cas “continu”; en particulier, c’est le cas de

[EGH99], ou un resultat de convergence des approximations par un schema de volumes finis

est demontre pour l’equation de Richards, qui contient (pL) au cas scalaire pour p = 2 .

On a choisi l’approche inverse, c’est-a-dire, d’appliquer les arguments du cas continu aux

30 Introduction

versions “continues” de la solution discrete et du systeme d’equations algebriques qui la

determine. Le premier pas consiste donc a ecrire ce systeme sous la forme d’une equation

dans D′ . On arrive ensuite a garder la carcasse de la demonstration de la convergence, et

de ramener completement cette question de la convergence a la question de la consistance de

l’approximation de l’operateur elliptique par le schema choisi. La demonstration de la consis-

tance exige des nouveaux arguments, propres au cas discret, et impose des restrictions sur la

classe de schemas qu’on considere. Cela concerne tout particulierement le choix des moyens

d’approximation de la composante tangentielle du gradient de la solution aux interfaces des

volumes de controle. Ce probleme n’a pas de solution generalement admise; de plus, celles

que je connaıs ne sont pas suffisamment exactes pour nos besoins. On propose un exemple

d’approximation qui verifie nos hypotheses. Il faut pourtant indiquer que ces hypotheses, bien

qu’elles rendent la demonstration de la consistance plutot elegante, paraissent trop limitatives

du point de vue numerique.

5. A la fin, je citerai quelques-uns des problemes ouverts que laisse entrevoir ce memoire.

L’unicite d’une solution entropique generalisee pour une loi de conservation scalaire multidi-

mensionnelle avec une fonction de flux continue dans L1 ∩ L∞ demeure inconnue.

La pertinence des conditions de structure de type a(v,Dv) = a(b(v), Dv) pour l’existence

d’une solution faible au probleme (Pr) me paraıt tres douteuse. Cependant, je n’ai pas

reussi a demontrer qu’on peut s’en passer et avoir toutefois la convergence forte d’une suite

des solutions aux problemes approches, ne serait ce que dans le cadre de l’equation modele

b(v)t = vxx + F (v)x ([BW99]) ou l’existence est connue.

Le “chain rule” lemme (ChR) , qui est indispensable dans le chapitre 2.I, reste a demontrer

dans sa formulation naturelle. Cela permettrait d’avoir facilement un resultat plus general

dans le cas des conditions au bord de Dirichlet non-homogenes.

La possibilite de constuire une theorie consistante de solutions non-classiques de certaines

lois de conservation ([LF98]) souleve des questions interessantes, de par sa relation avec les

effets dissipatifs-dispersifs (que l’on peut sans doute prendre auto-similaires) d’une part, et

de par son interpretation eventuelle dans le cadre de la theorie des semi-groupes non-lineaires

d’autre part.

L’etude precise du probleme de Riemann pour des systemes speciaux, par les methodes de

viscosite auto-similaire, ne s’arrete peut-etre pas aux systemes de la dynamique des gaz.

Meme en presence des resultats generaux sur les systemes 2×2 ([DDp76]), on peut esperer

d’obtenir plus d’information, y compris des “Riemann solvers” plus ou moins explicites, dans

des cas tres particuliers. Le cas echeant, cela peut englober des systemes qui ne sont pas

hyperboliques ou qui donnent naissance aux ondes specifiques.

Le resultat de convergence des schemas de volumes finis qu’on propose me semble amu-

sant du point de vue purement mathematique; mais il est douteux qu’il soit accepte par la


communaute des numericiens tant qu’il n’est pas etendu a des systemes plus realistes et a des

schemas moins contraignants. En particulier, il semble ([Ey..]) que le choix d’approximation

de la composante tangentielle du gradient sur les interfaces n’est pas tres important pour la

convergence des methodes de volumes finis. D’autre part, il serait interessant de voir ce que

pourrait apporter l’interpretation “continue” des schemas de volumes finis a l’etude d’autres

types de problemes.

Je continuerai a chercher des reponses a ces questions.

. . . no eto uge sovsem drugaja istorija.

En concluant cette introduction, je tiens a exprimer ma profonde reconnaissance

envers Stanislav Nikolaevıch Kruzhkov, qui m’a initie aux mathematiques, et je tiens

a remercier Philippe Benilan pour tout le soutien, d’ordre scientifique et d’ordre

humain, qu’il m’a apporte tout au long de notre connaissance, et qui m’a permis de

venir a bout de ce travail.

Part 1

Conservation Laws

with Continuous Flux Function

CHAPTER 1.I

The Riemann Problem

for Scalar Conservation Law

with Continuous Flux Function:

the Self-Similar Viscosity Approach†

Introduction

Consider the following Cauchy problem:

(1ε) Ut + f(U)x = εtUxx,

where f : IR → IR is continuous, U maps Π+ = IR+ × IR in IR , and ε ≥ 0 ;

(2) U(0, x) = U0(x) =

u−, x < 0

u+, x > 0

For the sake of simplicity, assume u− < u+ .

Our main concern is the Riemann problem for the scalar conservation law (10) . It is

well known that, because of the non-uniqueness of a weak solution to (10), (2) , additional

criteria have to be introduced in order to select the admissible one. We seek to establish the

uniqueness of a solution to (10), (2) satisfying the wave fan admissibility criterion, proposed

by C.M.Dafermos in [D89] in the context of general hyperbolic systems of conservation laws.

More exactly, we seek to distinguish the (wave-fan admissible) solution to (10), (2) as the

unique a.e. limit of solutions to (1ε), (2) as ε ↓ 0 . From this viewpoint, the term εtUxx

introduces vanishing artificial viscosity in (10), (2) , which we will call the self-similar viscosity.

A related approach to the admissibility for the Riemann problem has earlier been pursued

by A.S.Kalashnikov in [Ka59] (for the scalar case), V.A.Tupchiev in ([Tu64, Tu66, Tu73])

†This chapter is an extended English version of the note [BA2], which refines the approach used in [BA1].

36 Riemann problem for Scalar Conservation Laws

and C.M.Dafermos in [D73a, D74] (for the case of systems), and many others since then

(cf. the survey paper [Tz98]). The idea is to establish the existence and a uniform BV

bound for solutions Uε , ε > 0 ; pass to the limit by the compactness argument; derive

self-contained pointwise conditions on discontinuities of a limiting function, which is a weak

solution of (10), (2) ; and then infer the uniqueness when possible. In this way, the problem

under consideration has already been solved by A.S.Kalashnikov in [Ka59]. In the smooth

case, he had shown that any limit of solutions of (1εn), (2) as εn ↓ 0 fulfill conditions

that assure uniqueness for the Riemann problem (10), (2) (cf. (9) in Remark 3 at the end

of the chapter). As it is shown in [D73a], the most general form of such conditions is the

entropy-entropy flux admissibility (cf. [K70a, Lax71]).

For the case of scalar conservation law with continuous flux function, the wave-fan ad-

missibility for the Riemann problem is still equivalent to the entropy admissibility as defined

by S.N.Kruzhkov (see the notion of generalized entropy solution in Definition 1 below) in

[K69a, K69b, K70a] (cf. also A.I.Vol’pert, [V67]). This is due to the fact that there is unique-

ness of a generalized entropy solution to the Cauchy problem for (10) with general initial data

in L∞ even in the case the flux function is only continuous. Recall the following classical

definition.

Definition 1 A bounded measurable in Π+ function U(·, ·) is a generalized entropy solution

to the problem (10), (2) , if

(i) for all k ∈ IR , ψ ∈ C∞0 (Π+) such that ψ ≥ 0 one has

∫ ∫

Π+

|U(t, x)− k|ψt(t, x) + sign [U(t, x)− k][f(U(t, x))− f(k)]ψx(t, x) dxdt ≥ 0;

(ii) there exists a set E ⊂ IR+ of measure |E| = 0 such that for all t ∈ IR+ \ E the

function U(t, ·) is defined a.e. on IR , and for all r > 0 one has

limt∈R+\E, t→0

∫

|x|≤r

|U(x, t)− U0(x)|dx = 0.

While the general theory of the Cauchy problem for scalar conservation laws with Lipschitz

continuous flux function is due to S.N.Kruzhkov ([K69a, K69b, K70a]), results in the case of

only continuous flux function f(·) were first established by Ph.Benilan ([B72]) by the nonlinear

semigroup approach, which yields the existence and uniqueness for L1 initial data in the one-

dimensional case. The uniqueness of a generalized entropy solution for the case including the

Riemann problem for one-dimensional scalar conservation law with continuous flux function has

first been established by S.N.Kruzhkov and P.A.Andreyanov ([KPA75]). For general L∞ initial

data, the uniqueness in this case has been shown by L.Barthelemy ([Ba88]) and S.N.Kruzhkov,

E.Yu.Panov ([KP90]). Note that in the multidimensional case, the uniqueness is false in general

for L∞ data (cf. [KP90]) and still not clear for L1 ∩ L∞ data (cf. [BK96]). Some further

1.I.0. Introduction 37

results in this last direction are presented in Chapter 1.IV and [BABK]; while they have recently

been extended by E.Yu.Panov ([P99, P..]), the problem remains open.

It follows that for (10), (2) with only continuous f the approach of [Ka59] still could be

used. Arguing as in [K70a, K70b], one easily shows that a wave-fan admissible weak solution

of (10), (2) , whenever it exists, is a solution in the sense of Definition 1, so that it is the

unique generalized entropy solution. Note that, a fortiori, it is the unique solution to the

Riemann problem (10), (2) in the class of all wave-fan admissible solutions.

Here we bypass both the compactness and the entropy admissibility arguments, and prove

the uniqueness (and existence) of a wave-fan admissible weak solution independently, using

the somewhat elementary structure of the problem (10), (2) . The approach is essentially

one-dimensional (with respect to U as well as with respect to x ). Nevertheless, it can be

extended to special systems of conservation laws that have a kind of scalar structure (for the

system of isentropic gas dynamics, this is done in Chapters 1.II, 1.III herein) and where a

uniqueness result of general authority seems to be unknown.

The results of this chapter are summarized in Theorems 1,2 in Section 2. They state

that for all ε > 0 there exists a unique bounded self-similar solution Uε to (1ε), (2) ,

and Uε converge as viscosity vanishes. The limiting function U , which is the wave-fan

admissible solution to (10), (2) , is given by an explicit formula. Curiously, we do not need

the a priori knowledge of this formula for U . Indeed, the starting point for our approach is

the observation, going back at least as far as the lecture notes [G59] of I.M.Gelfand, on the

profile of distribution solutions to (10), (2) obtained through approximation of shock waves

by travelling waves. It can be summarized by saying that (in case u− < u+ ) the admissible

solution of (10), (2) is the graph inverse function to the derivative of the convex hull of f(·)on [u−, u+]

†, or else by the formula

(3) U(x/t) = ∂/∂x maxu−≤v≤u+

(xv − tf(v)).

These two assertions are actually equivalent, due to the Fenchel formula. A careful construction

of the “travelling waves”-admissible solution in case of smooth f(·) can be found in lecture

notes [K70b] by S.N.Kruzhkov (see also [GR], and [ChHs] and references therein); in [GiNTe],

the profile of the solution is directly obtained from Definition 1. Writing U under the form

chosen above permits to compare the formula (3) with formulae, suggested by the convex

analysis, that were proposed in [H50, Lax57, KPe87] for different f(·) and U0(·) . Due to

the uniqueness of a generalized entropy solution, they are all equivalent in the case of Riemann

problem with regular convex flux function.

In the lengthy Remark 3 at the end of the chapter some comments on interrelations of

the entropy admissibility, usual vanishing viscosity, self-similar viscosity and approximation by

†We refer to the greatest convex function F (·) on [u−, u+] such that F ≤ f as to the convex hull of

f(·) on [u−, u+] .

travelling waves are presented.

1 Restatement of the problem

First let ε > 0 be fixed. Let restrict our attention to bounded self-similar solutions of

(1ε), (2) . For simplicity, we assume in the sequel that u− < u+ . Integrating the differential

equation for U(ξ) = U(t, x) , where ξ = x/t , we arrive to the following definition.

Definition 2 Let ε > 0 . A solution of (1ε), (2) is a function U(t, x) = U(x/t) , U ∈C1(IR) , that verifies

(4) εU ′(ξ) = −∫ ξ

ξ0

ζU ′(ζ)dζ + f(U(ξ)) +K with some ξ0, K ∈ IR, and U(±∞) = u±.

The strict monotony property for solutions of (4) , trivial in the case of smooth flux

function, can fail if f(·) is only continuous (cf. Remark 2). Nevertheless, we have the

following result.

Lemma 1 All solution U(ξ) of the problem (4) is non-decreasing on IR .

Proof: Assume the contrary. Then there exists c a point of extremum of U(·) on IR such

that U(c) = u0 6= u± . For definiteness, assume that c is a point of local maximum. Consider

the greatest segment [c1, c2] containing c such that U |[c1,c2] ≡ u0 . For all α > 0 small

enough there exist ξ1 = ξ1(α) , ξ2 = ξ2(α) such that ξ1 < c1 ≤ c2 < ξ2 , U(ξ1) = U(ξ2) =

u0 − α , and the distance between ξ2 and ξ1 is the least possible. Since U ∈ C1(IR) ,

there exists maxξ∈[ξ1,ξ2] |U ′(ξ)| = |U ′(ξ)| =M > 0 with some ξ ∈ [ξ1, ξ2] . For definiteness,

assume that c2 < ξ ≤ ξ2 Consider separately the two possibilities.

a) U ′(ξ) = −M . In this case U(ξ1) = U(ξ2) ≤ U(ξ) < u0 = U(c1) . Take ξ2 = ξ

and ξ1 = maxξ| ξ1 ≤ ξ < c1,U(ξ) = U(ξ) . One has U > U(ξ1) on [ξ1, c1] , therefore

U ′(ξ1) = m ≥ 0 . Taking into account that U ′|[c1,c2] ≡ 0 , and using (4) with ξ0 = c and

the corresponding constant K = Kc , one obtains

εU ′(ξ1) = −∫ ξ1

c1

ζU ′(ζ)dζ + f(U(ξ1)) +Kc,

εU ′(ξ2) = −∫ ξ2

c2

ζU ′(ζ)dζ + f(U(ξ2)) +Kc.

Subtracting these two equalities, one finds that ε(M +m) ≤ M · (|ξ1| + |ξ2|) · ((c1 − ξ1) +

(ξ2 − c2)) .

b) U ′(ξ) =M > 0 . According to the choice of ξ2 , one has ξ < ξ2 and U(ξ) > U(ξ2) , so

that there exists ξ2 = minξ| ξ < ξ < ξ2,U(ξ) = U(ξ) . It follows that U ′(ξ2) = −m ≤ 0 .

Set ξ1 = ξ and argue as in case a).

1.I.1. Restatment of the problem 39

In the two cases, we infer that ε ≤ (|ξ1|+ |ξ2|) · ((c1 − ξ1) + (ξ2 − c2)) . As α → +0 , one

has ξ1 → c1 , ξ2 → c2 , so that ε ≤ 0 at the limit. This contradiction proves the lemma.

⋄

By Lemma 1, the function Ξ(u) = [U(ξ)]−1 is defined a.e. on [u−, u+] and monotone.

Proposition 1 Assume that a function U is a solution of (1ε), (2) in the sense (4) . Then

the function Φ(u) =

∫ u

U(0)

Ξ(v)dv −K on [u−, u+] is a solution of the following problem :

(5)

Φ ∈ C[u−, u+] and Φ is convex

Φ(u) ≤ f(u) on [u−, u+] and Φ(u±) = f(u±)ε

f−Φ∈ L

loc1 (u−, u+)

Φ(u) ≥ εf(u)−Φ(u)

(f(u)− Φ(u))(Φ(u)− ε

f(u)−Φ(u)

)= 0

in the sense of measures

on (u−, u+)

(in the rest of the chapter, ˙ stands for d/du ).

Conversely, assume Φ(·) is a solution of (5) . Then the function

U(t, x) = U(x/t) = [Φ(u)]−1 ≡ ∂/∂x maxu−≤v≤u+

(xv − tΦ(v))

is a solution of (1ε), (2) in the sense (4) .

Proof: (4) ⇒ (5) Rewrite the equation (4) under the form

(6) εU ′(ξ) = f(U(ξ))− Φ(U(ξ));

it follows that Φ ∈ C(u−, u+) and Φ ≤ f . Since Φ(u) = Ξ(u) is an a.e. continuous,

non-decreasing function, Φ is convex. Set Ω = u| ∃ξ : U(ξ) = u, U ′(ξ) = 0 . The

Lebesgue measure |Ω| is zero, by the Sard lemma, and Ω ≡ u|Φ(u) = f(u) . For all

u ∈ [u−, u+] \ Ω there exists Φ(u) = Ξ(u) = 1/U ′(ξ) > 0 ; we have Φ(u) = εf(u)−Φ(u)

and Φ(u) < f(u) . Since (f − Φ)Φ = 0 on Ω , it follows that (f − Φ)(Φ− ε

f−Φ

)= 0

in the sense of measures on (u−, u+) . Since |Ω| = 0 and Φ ≥ 0 , one has Φ ≥ εf−Φ

on (u−, u+) in the same sense. Consequently, for all segment [a, b] ⊂ (u−, u+) one has∫ b

aε

f(u)−Φ(u)du ≤ Φ(b + 0) − Φ(a − 0) < ∞ , so that ε

f−Φ∈ L

loc1 (u−, u+) . The equation

(6) , together with U(±∞) = u± , imply that the limits limξ→±∞

U ′(ξ) exist and are zero, so

that Φ(u±) = f(u±) and Φ ∈ C[u−, u+] .

(5) ⇒ (4) Define the multivalued function Ξ(·) by Ξ : u ∈ [u−, u+] 7→ [Φ(u−0), Φ(u+

0)] , with Φ(u± ± 0) = ±∞ . Set Ω := u|Φ(u) = f(u) . It is clear that Ξ is strictly

increasing and single-valued on the complementary of Ω . Let U = [Ξ]−1 ; one has U ∈C(IR) . If u0 /∈ Ω , there exists Ξ(u) = ε

f(u)−Φ(u)> 0 in a neighbourhood of u0 , so that

(6) is satisfied at the point ξ0 = Ξ(u0) . If u0 ∈ Ω and ξ0 ∈ Ξ(u0) , then for all α > 0

there exists a neighbourhood of u0 such that f(u)− Φ(u) < εα ; consequently, Φ > 1/α

in this neighbourhood. It follows that |Ξ(u0 + δ) − ξ0| ≥ |δ|/α for all δ small enough;

therefore there exists U ′(ξ0) = 0 and (6) is satisfied in all the cases. Thus U ∈ C1(IR)

and (6) implies (4) , by the definition of U(·) . Clearly, U(±∞) = u± . By the Fenchel

formula, it follows that the other representation of U holds. ⋄

Remark 1 It is interesting to observe that , according to Proposition 1, the function w(t, x) =

Φ(U(t, x)) satisfies the Hopf equation ([H50]) wt +wwx = 0 if U is a solution of (1ε), (2)

in the sense (4) .

2 Results and proofs

In this section we investigate solvability and convergence properties for the problem (5) and

deduce the corresponding results for the Riemann problems (1ε), (2) .

Proposition 2 For all ε > 0 , u− < u+ and f ∈ C[u−, u+] there exists a unique solution

to the problem (5) .

Proof: Let argue by reductio ad absurdum in order to prove the uniqueness of a solution to

(5) . Assume Φ,Ψ are two different solutions, and c be a point of (positive) local maximum

of (Φ − Ψ) on [u−, u+] . In fact, in case Φ(c) < f(c) one could get a contradiction

from the standard maximum principle. In the general case, one can find ∆ > 0 such that

[c, c + ∆] ⊂ [c, b) and Φ ≥ εf−Φ

> εf−Ψ

= Ψ in the sense of measures on [c, c + ∆] .

Indeed, one has Ψ(c) < Φ(c) ≤ f(c) , so that it can be assumed that Ψ ∈ C2[c, c+∆] and

satisfies (5) with equality on [c, c + ∆] . Moreover, the assumption of maximality above,

together with the convexity of Φ , imply that there exists Φ(c) = Ψ(c) . It follows that

Φ(c+ δ ± 0) > Ψ(c+ δ ± 0) for all δ ∈ (0,∆) , which contradicts to the choice of c .

In order to prove the existence, introduce the penalized problem∗

(7)

Φn(u) = Gn(u,Φn(u)) =

εf(u)−Φn(u)

∧ n, n = 1, 2, ...

Φn(u±) = f(u±), Φn ∈ C2[u−, u+].

Since Gn(u,Φ) is continuous on u and Φ is bounded, there exists a solution of (7) .

The maximum principle is verified for equations of type (7) , because Gn(u,Φ) is in-

creasing in Φ . Set G =√ε(u− u−)(u+ − u) and denote by F the convex hull of f on

[u−, u+] . One has (F − G) ≥ −G ≥ εG≥ ε

f−(F−G), so that (F − G) is a subsolution of

the problem (7) corresponding to G∞ = εf(u)−Φ

. One finds for n ≥ m that G∞(u,Φ) ≥≥ Gn(u,Φ) ≥ Gm(u,Φ) ; by the maximum principle, it follows that Φm(u) ≥ Φn(u) ≥≥ F (u)−G(u) on [u−, u+] .

∗Let a ∧ b denote minb,maxa, 0 for a, b ∈ IR .

1.I.2. Results and proofs 41

Thus Φn(u) ↓ Φ(u) ∈ IR for all u ∈ [u−, u+] ; in addition, Φ(u±∓0) = Φ(u±) = f(u±)

so that Φ ∈ C[u−, u+] and Φ is convex. Moreover, Gn(u,Φn(u)) tends to εf(u)−Φ(u)

∈ IR+

on [u−, u+] . Take a test function ϕ ∈ C∞0 (u−, u+) such that ϕ ≥ 0 ; by (7) and the

Fatou lemma, one has

(8)

u+∫

u−

ϕΦ(u) du = limn→∞

u+∫

u−

ϕΦn(u) du =

= limn→∞

u+∫

u−

ϕGn(u,Φn(u)) du ≥u+∫

u−

ϕε

f(u)− Φ(u)du.

Therefore Φ ≥ εf−Φ

in the sense of measures on (u−, u+) , andε

f−Φ∈ L

loc1 (u−, u+) .

Now take a test function ϕ ∈ C∞0 (u−, u+) with suppϕ ⊂ u|Φ < f . In this

case (8) becomes an equality, because there exists N = N(ϕ) such that for all n ≥ N ,

Gn(u,Φn(u)) ≤ GN (u,ΦN(u)) on suppϕ . Since f − Φ = 0 on [u−, u+] \ u|Φ < f ,one has (f − Φ)

(Φ− ε

f−Φ

)= 0 in the sense of measures on (u−, u+) . ⋄

Remark 2 It is easy to show that for a Lipschitz continuous flux function f the solution of

(5) is a classical solution to the equation Φ = εf−Φ

on (u−, u+) . Nevertheless, it can be

shown using the maximum principle that for f(u) =√u and for all interval (u−, u+) ∋ 0

sufficiently small, the derivative of the solution to (5) has a positive jump at 0 . This jump

corresponds to an interval of ξ where the solution U(·) of (1ε), (2) is constant.

Propositions 1 and 2 yield the following result.

Theorem 1 For all ε > 0 , u− < u+ and f ∈ C[u−, u+] there exists a unique solution Uε

to the problem (1ε), (2) in the sense of Definition 2. This solution is given by the formula

analogous to (3) :

Uε(t, x) = Uε(x/t) = ∂/∂x maxu−≤v≤u+

(xv − tΦε(v)),

where Φε(·) is the unique solution of (5) .

Since we are interested in passing to the limit as ε → 0 , let introduce the subscript ε in

the notation for solutions of (4) and (5) . We need the following two lemmae.

Lemma 2 Let F (·) be the convex hull of f(·) on [u−, u+] . Then Φε(·) converge to F (·)uniformly on [u−, u+] as ε → +0 .

Proof: The proof is based upon a kind of maximum principle argument.

First note that Φε ≤ F . On the other hand, for all α > 0 there exists a function

G ∈ C2[u−, u+] such that 0<F−G<α and G≥C(α)>0 on [u−, u+] . Let c(ε, α) be a

point of global maximum of G−Φε on [u−, u+] ; assume that this maximum is positive. Then

Φε(c) ≤ G(c) < F (c) ≤ f(c) , so that there exists Φε(c) ≥ G(c) ≥ C(α) . Consequently,ε

f(c)−Φε(c)≥ C(α) , and for all ε small enough one has G−Φε < α on [u−, u+] . It follows

that F − Φε < 2α on [u−, u+] in this case; besides, this last inequality is evident in case

the maximum is nonpositive, whence the claim of the lemma. ⋄

Lemma 3 Let F n(·) , n = 1, 2, ... be a sequence of convex functions converging to F 0(·)on [u−, u+] . Then the sequence Un(·) tends to U0(·) at the points ξ of continuity of

U0(·) , where Un(·) , n = 0, 1, ... are the functions constructed by the formula (3) applied

to the functions F n .

Lemma 3 is a corollary of the Fenchel formula and general theorems of the convex analysis

and basic probability theory. An elementary proof can be found in [BA1].

Finally, we establish the relation between the problem (10), (2) and the self-similar viscosity

regularized problems (1ε), (2) , ε > 0 .

Theorem 2 Let Uε be the solution to the problem (1ε), (2) in the sense of Definition 2

(which exists and is unique, due to Theorem 1). Then Uε converge a.e. on Π+ to the

function

U(t, x) = U(x/t) = ∂/∂x maxu−≤v≤u+

(xv − tF (v)) ≡ ∂/∂x maxu−≤v≤u+

(xv − tf(v))

as ε→ +0 , and U is a generalized entropy solution of the problem (10), (2) .

Proof: The convergence of Uε to U follows readily from Lemmae 2,3 and the Fenchel

formula. It is easy to see that U(t, x) → u± as x/t → ±∞ , which implies that (i) of

Definition 1 holds. Moreover, U is a limit of viscous approximations of (10), (2) (i.e., a

wave-fan admissible weak solution). Using Kruzhkov’s techniques (cf. [K69a, K69b, K70a]),

one easily deduces that U satisfies (i) of Definition 1 as well. ⋄

Remark 3 As it is underlined in [Tz96], the wave fan admissibility criterion is different from

usual admissibility conditions for the Riemann problem in that it penalizes the whole fan of

shocks and rarefactions in the solution and not the shocks one by one. In this sense, it is

closer to the global criteria such as the Kruzhkov entropy admissibility criterion (Definition 1)

or the Dafermos entropy rate admissibility criterion (cf. [D73b]).

In piecewise continuous solutions of scalar conservation laws with continuous flux function,

the Kruzhkov criterion still decides on admissibility of each shock in a solution separately; it

induces the conditions

(9) S(u) := sign (ur−ul)s(u−ul)− (f(u)−f(ul)) ≤ 0 for all u between ul and ur,

and S(ur) = 0 if the shock joins ul at the left to ur at the right and propagates with

the speed s . This property is directly motivated by the wave fan admissibility criterion for

1.I.2. Results and proofs 43

the Riemann problem, on account of the explicit representation for the solution. At the same

time, motivation by the travelling waves condition, which is the most common outcome of the

self-similar viscosity approach, becomes more delicate when f is not Lipschitz continuous.

Indeed, this condition requires that an admissible shock could be approximated by solutions

Uε to the equation

(10) Ut + f(U)x = εUxx

of the form Uε(t, x) = Uε((x − st)/ε) † (in case of multiple inflexion points, one admits

also that it be a limit of such shocks with respect to infinitesimal perturbations of f or/and

ul, ur ). Such approximation is possible if and only if there exists a solution to the problem

(11)d

dζU(ζ) = S(U(ζ)), U(−∞) = ul, U(+∞) = ur.

In case f is Lipschitz continuous, this yields the condition (9) with strict inequality, by virtue

of uniqueness for the ODE Cauchy problem. In case of f continuous, analysis of stationary

points of the equation (11) shows that one still can deduce (9) from the travelling waves

condition.

The relation between the usual and self-similar viscosity limits for the Riemann problem

(10), (2) becomes less clear in absence of regularity of f . Kalashnikov in [Ka59] proves the

equivalence, starting from the uniqueness of a self-similar viscous limit and using the maximum

principle to compare primitives of solutions to (1ε) and (10) . In absence of regularity of

f , this comparison becomes delicate†. A reason for the equivalence remains the uniqueness

theory for generalized entropy solutions (e.g., cf. [KPA75] and Chapter 1.IV).

In conclusion, the wave fan admissibility criterion directly selects a unique solution to the

Riemann problem for a scalar conservation law with continuous flux function. As it is shown

in Chapters 1.II and 1.III, the same is true for the p-system and the corresponding system in

Eulerian coordinates, even in the case vaccuum appears; for the p-system, the hyperbolicity

condition can be omitted.

†Clearly, both travelling waves and self-similar viscosity approaches to the Riemann problem are motivated

by introducing the vanishing viscosity of type (10), which is the original idea came from the fluid mechanics

(cf. [Ray10]). Even if the rigorous mathematical study of discontinuous solutions of conservation laws had

started from such a description for Burgers’ equation, given by E.Hopf in his pioneering work [H50], the limits

of solutions of (10) remain difficult to describe directly, especially in case of systems. The recent success in

proving uniqueness for the Glimm scheme and the wave-tracking algorithm, due to A.Bressan and collaborators

(cf. [Br99] for a survey of results), does not promise the uniqueness of solutions that are limits of vanishing

viscosity unless uniform BV estimates on viscous solutions are obtained.

On the other hand, approximation by travelling waves or self-similar viscosity permits to pursue the analysis

within the field of ordinary differential equations, where a variety of tools apply. From the physical viewpoint,

both seem acceptable but not natural.

†Maximum principle is an argument of the same order as the monotony of Uε in Lemma 1. While the

assertion of Lemma 1 is evident when f is smooth, the author was unable to establish it in the general case

by a simpler method than that exposed in the proof.

CHAPTER 1.II

The Riemann Problem for p-System

with Continuous Flux Function†

Introduction

Consider the Riemann problem for a so-called p-system, i.e. the initial-value problemUt − Vx = 0

Vt − f(U)x = 0, (U, V ) : (t, x) ∈ IR+ × IR 7→ IR2; (1)

U(0, x) =

u+, x > 0

u−, x < 0, V (0, x) =

v+, x > 0

v−, x < 0u±, v± ∈ IR. (2)

The flux function f : IR 7→ IR is assumed to be continuous and strictly increasing (except in

Section 5, where the monotony assumption is relaxed).

In the case of piecewise smooth flux function the problem (1),(2) was treated by L.Leibovich,

[Le74] (cf. also [ChHs] and references therein). By analyzing the wave curves on the plane

(u, v) it has been shown that a self-similar distribution solution that is consistent with a

certain admissibility criterion (cf. B.Wendroff, [We72]; also I.Gelfand, [G59] and S.Kruzhkov,

[K70b] for the original idea carried out in the case of scalar conservation laws) may be explicitly

constructed through convex and concave hulls of the flux function f . It has been noticed by

C.Dafermos in [D74] that the same solution satisfies the wave fan admissibility criterion, i.e.,

it can be obtained as limit of self-similar viscous approximations as viscosity goes to 0 . Here

we follow this last idea.

Let introduce some notation. For given [a, b] ⊂ IR and f : u ∈ [a, b] 7→ IR continuous,

the convex hull of f on [a, b] is the function u ∈ [a, b] 7→ supφ(u) | φ is convex and φ ≤

†This chapter extends the author’s graduate paper [BA0] written at the Chair of Differential Equations

under the supervision of S.N.Kruzhkov. The contents of this chapter, excluding Sections 4 and 5, will be

published in [BA3].

46 Riemann problem for p-Systems

f on [a, b]. Respectively, the concave hull of f on [a, b] is the function u ∈ [a, b] 7→

infφ(u) | φ is concave and φ ≥ f on [a, b]

. Take u0 in IR ; by F+(·; u0) denote the

convex hull of f on [u0, u+] if u0 ≤ u+ , and the concave hull of f on [u+, u0] if

u0 ≥ u+ . Replacing u+ by u− , define F−(·; u0) in the same way. Let shorten F±(·; u0)to F± when no confusion can arise.

Since f is strictly increasing, the inverse of dF+

du, denoted by

[dF+

du

]−1

, is well defined in

the graph sense as function from [0,+∞) to [u0, u+] if u0 < u+ (respectively, to [u+, u0]

if u0 > u+ ). In the case u0 = u+ let[dF+

du

]−1

mean the function on [0,+∞) identically

equal to u0 . With the same notation for F−, u− in place of F+, u+ and F± standing fordF±

du, which are non-negative, the self-similar solution of the problem (1),(2) constructed in

[Le74] may be written as

U(t, x) =

[F+(·; u0)

]−1

(x2/t2), x ≥ 0[F−(·; u0)

]−1

(x2/t2), x ≤ 0, (3)

V (t, x) = v− −∫ x/t

−∞

ζdU(ζ), (4)

dU(ζ) being regarded as measure; and, for a bijective flux function f , the value u0 is

uniquely determined by

v− − v+ =

∫ u+

u0

√F+(u; u0)du+

∫ u−

u0

√F−(u; u0)du. (5)

In the case of bijective locally Lipschitz continuous flux function f , the same formulae (3)-

(5) were obtained by P.Krejcı, I.Straskraba ([KrSt93]) for the unique solution to satisfy their

“maximal dissipation” condition. This solution was also shown to be the unique a.e-limit as

ε→ 0 of solutions to Riemann problem for the p-system regularized by means of infinitesimal

parameter ε > 0 , introduced into the flux function f , and the viscosity

(0

εtVxx

).

In this chapter a refinement of these results is presented. The techniques employed are

those used by the author while treating the Riemann problem for a scalar conservation law with

continuous flux function (cf. Chapter 1.I and [BA1],[BA2]). In the general case of continuous

strictly increasing flux function f , the Riemann problem (2) for the p-system (1) and the

regularized system

Ut − Vx = 0

Vt − f(U)x = εtVxx(6)

are treated. The main result is the following theorem:

1.II.1. Restatement of the problem 47

Theorem 1 Suppose f : IR → IR is increasing and bijective. Then for all u±, v± ∈ IR ,

ε > 0 there exists a unique bounded self-similar distribution solution (Uε, V ε) of the problem

(6),(2).

Besides, as ε ↓ 0 , (Uε, V ε)(ξ) → (U, V )(ξ) a.e. on IR , where (U, V ) is given by the

formulae (3)-(5), so that (U, V ) is a self-similar distribution solution of the problem (1),(2).

The bijectivity condition is only needed for the existence of solutions and cannot be omitted

(see Remark 7.6 in [KrSt93]), though it can be relaxed (see Remark 2 in Section 3).

The chapter is organized as follows. In the first section the problem (6),(2) is reduced

to a pair of boundary-value problems for a second-order ordinary differential equation on the

domains (minu0, u±,maxu0, u±) ; u0 is a priori unknown and satisfies an additional

algebraic equation. In Section 2 existence, uniqueness and convergence (as ε → 0 ) results

are obtained for the ODE problem stated in Section 1, with u0 ∈ IR fixed. Then it is shown in

Section 3 that u0 is in fact uniquely determined by the flux function f , ε , and the Riemann

data u±, v± ; finally, Theorem 1 above is proved†.

1 Restatement of the problem

We start by fixing ε > 0 . Consider the problem (6),(2) in the class of bounded distribution

solutions (U, V ) of (6) such that (U, V )(t, ·) tends to (U, V )(0, ·) in L1loc(IR) × L1

loc(IR)

as t tends to +0 essentially. Moreover, since both the initial data (2) and the system (6)

are invariant under the transformations (t, x) → (kt, kx) with k in IR (here is the reason

to introduce the viscosity with factor t ), it is natural to seek for self-similar solutions, i.e.

(U, V ) depending solely on the ratio x/t . By abuse of notation, let write (U, V )(t, x) =

(U, V )(x/t) . Let ξ denote x/t and use U ′, V ′ for dU/dξ, dV/dξ and so on.

Lemma 1 A pair of bounded functions (U, V ) : ξ ∈ IR 7→ IR2 is a self-similar distribution

solution of (6),(2) if and only if U, V, ξU ′ and V ′ are continuous on IR , the equations

εξU ′(ξ) = −∫ ξ

0

ζ2U ′(ζ)dζ + f(U(ξ)) + C (7)

V (ξ) = −∫ ξ

0

ζU ′(ζ)dζ +K (8)

are fulfilled with some constants C,K, and also

U(±∞) = u±, V (±∞) = v±. (9)

Besides, there exist ξ± in IR± , ξ− ≤ ξ+ , such that U, V are strictly monotone on each

of (−∞, ξ−) , (ξ+,+∞) , with U ′ 6= 0 , and U, V are constant on (ξ−, ξ+) .

†In Section 4 we present some comments on the results obtained in Section 3. Section 5 is devoted to the

case where the monotonicity assumption on f is relaxed, which gives rise to a hyperbolic-elliptic problem.

Proof: Let (U, V ) be bounded self-similar distribution solution of the system (6). Then

−ξU ′−V ′ = 0 and −ξV ′−f(U)′ = εV ′′ in D′(IR) ; therefore[ξ2U−f(U)+εξU ′

]′= 2ξU

in D′(IR) . Since U ∈ L∞(IR) , it follows that

ξ2U − f(U) + εξU ′ =

∫ ξ

0

2ζU(ζ)dζ + C ∈ C(IR) (10)

with some C in IR . Hence one deduce consecutively that ξU ′ ∈ L∞loc(IR) , U ∈ C(IR\0)

and finally, U ∈ C1(IR\0) . Thus for all ξ 6= 0 (7) holds.

Now let prove the monotony property stated. For (ξ−, ξ+) take the largest interval in IR

containing ξ = 0 such that U = U(0) on (ξ−, ξ+) . For instance, let ξ+ be finite and

therefore U not constant on (0,+∞) ; suppose U is not strictly monotone on (ξ+,+∞) .

Since U ′ ∈ C(ξ+,+∞) , it follows that there exists c > ξ+ such that U ′(c) = 0 and

U ′ is non-zero in some left neighbourhood of c . For instance, assume U ′ > 0 in this

neighbourhood. Clearly, there exists a sequence ξn ⊂ IR increasing to c such that for all

n ∈ IN the maximum of U ′ on [ξn, c] is attained at the point ξn . Since f is increasing,

it follows that f(U(ξn)) < f(U(c)) . Take (7) at the points ξ = ξn and ξ = c ; subtraction

yields

εξnU′(ξn)− εc · 0 ≤

∫ c

ξn

ζ2U ′(ζ)dζ + f(U(ξn))− f(U(c)) ≤ U ′(ξn)

∫ c

ξn

ζ2dζ.

As n→ ∞ , one deduces that ε ≤ 0 , which is impossible.

Thus U , and consequently V , are indeed monotone on (−∞, 0) and (0,+∞) ; there-

fore there exist U(±0) = limξ→±0 U(ξ) . Hence by (10) there exist limξ→±0 ξU′(ξ) , which

are necessarily zero since U ∈ L∞(IR) . Thus (10) yields f(U(+0)) = f(U(−0)) , so that

U ∈ C(IR) . Consequently, ξU ′ ∈ C(IR) , V ′ ∈ C(IR) , and V ∈ C(IR) . It follows that

(7),(8) hold for all ξ in IR .

The converse assertion, i.e. that (7),(8) imply (6) in the distribution sense, is trivial.

Finally, since U and V are shown to be monotone on IR± whenever (7),(8) hold, it is

evident that (9) is fulfilled if and only if self-similar U, V satisfy (2) in L1loc -sense as t→ 0

essentially. ⋄

Let use this result to obtain another characterisation of self-similar solutions to (6),(2).

The idea is to seek for solutions of the same form as in formulae (3)-(5), substituting F± by

appropriate functions depending on ε . One thus has to “inverse” (3)-(5).

Set u0 := U(0) and consider (7) separately on (−∞, ξ−) , (ξ−, ξ+) , and (ξ+,+∞) ,

where ξ± are defined in Lemma 1. Assume u0 6= u− , u0 6= u+ . Let introduce the

notation I(a, b) for the interval between a and b in IR . One has U(ξ) = u0 for all

ξ ∈ (ξ−, ξ+) ; besides, the inverse functions U−1+ : I(u0, u+) 7→ (ξ+,+∞) and U−1

− :

1.II.1. Restatement of the problem 49

I(u0, u−) 7→ (−∞, ξ−) are well defined. For all u ∈ I(u0, u+) (respectively, u ∈ I(u0, u−) )

set

Φε+(u; u0) :=

∫ u

u0

(U−1+ (w)

)2dw − C

(resp., Φε

−(u; u0) :=

∫ u

u0

(U−1− (w)

)2dw − C

)(11)

with C taken from (7). The shortened notation Φ±(u) will be used for Φε±(u; u0) whenever

ε, u0 are fixed. Now (7) can be rewritten as εξU ′(ξ) = f(U(ξ)) − Φ±(U(ξ)) for ξ ∈I(ξ±,±∞) . The reasoning in the proof of Lemma 1 shows that U is not only monotone,

but also U ′ is different from 0 outside of [ξ−, ξ+] . It follows that for all u in I(a, b) ,

where a = u0 , b = u+ (resp., for all u in I(a, b) , where a = u0 , b = u− ), the function

Φ+ (resp., Φ− ) is twice differentiable and satisfies the equation

Φ(u) =2εΦ(u)

f(u)− Φ(u), with Φ(u) > 0 and Φ(u) · (b− a) > 0. (12)

Hence Φ+ < f ( Φ+ > f ) if u0 < u+ (if u0 > u+ ), and the same for Φ− , u− in place

of Φ+ , u+ .

Note that one can extend the functions Φ+ , Φ− to be continuous on I(u0, u+) , I(u0, u−)

respectively, and in this case one has

Φ+(u0) = f(u0), Φ+(u+) = f(u+)(resp., Φ−(u0) = f(u0), Φ−(u−) = f(u−)

). (13)

Indeed, one gets Φ±(u0) = f(u0) directly from (11) and (7). Besides, for ξ ∈ IR± , εξU ′(ξ)

is equal to f(U(ξ))−Φ±(U(ξ)) , which has finite limits as ξ → ±∞ because U(±∞) = u±

and Φ± are convex and bounded on I(u0, u±) . The limits of εξU ′(ξ) cannot be non-zero

since U is bounded, thus one naturally assign Φ±(u±) := f(u±) .

Now from (8)-(11) it follows that

v− − v+ =

∫ u+

u0

√Φε

+(u; u0)du+

∫ u−

u0

√Φε

−(u; u0)du. (14)

Note that in the case u0 = u+ ( u0 = u− ), (12)-(14) formally make sense, with Φ+

defined at u = u0 = u+ by f(u+) (resp., with Φ− defined at u = u0 = u− by f(u−) ).

Finally, the reasoning above is inversible. More presisely, for given u0 ∈ IR and Φε±(·; u0) ∈

C2(I(u0, u±)) ∩ C(I(u0, u±)) such that (12)-(14) hold, define U, V by

U(ξ) =

[Φε

+(·; u0)]−1

(ξ2), ξ ≥ 0[Φε

−(·; u0)]−1

(ξ2), ξ ≤ 0(15)

V (ξ) = v− −∫ ξ

−∞

ζdU(ζ), (16)

with [Φε+(·; u0)]−1 (and [Φε

−(·; u0)]−1 ) taken in the graph sense and equal to u+ (to u− )

identically whenever u0 = u+ ( u0 = u− ). Then (U, V ) satisfy (7)-(9). Indeed, U is

continuous, Φε+(u0; u0) = Φε

−(u0; u0) , and the equation εξU ′(ξ) = f(U(ξ))−Φε±(U(ξ); u0)

holds for all ξ ∈ IR± . Hence ξU ′ ∈ C(IR) and (7) is true. Therefore V ′, V are continuous

and (8),(9) are easily checked.

We collect the results obtained above in the following proposition:

Proposition 1 Let ε, f, u±, v± be fixed. Formulae (15),(16) provide a one-to-one corre-

spondence between the sets A and B defined by

A :=(u0,Φ±(·)

)| u0 ∈ IR, Φ± : I(u0, u±) 7→ IR, Φ± ∈ C2(I(u0, u±)) ∩ C(I(u0, u±))

and (12)− (14) hold

B :=(U, V ) | (U, V ) is a bounded self-similar distribution solution of (6), (2)

In fact, it will be shown in Section 3 that A and thus B are one-element or empty sets.

The resemblance of formulae (3),(4),(5) and (15),(16),(14) permits to get the convergence

result of Theorem 1 if one has convergence of Φε± to F± as ε → 0 .

2 The problem (12),(13) with fixed domain

Let fix a, b ∈ IR and consider the equation (12) on the interval I(a, b) , with the boundary

conditions as in (13). For instance, suppose a ≤ b .

Proposition 2 For all continuous strictly increasing f , ε > 0 , and a, b ∈ IR there exists a

unique Φ in C2(I(a, b)) ∩ C(I(a, b)) satisfying (12) such that Φ(a) = f(a) and Φ(b) =

f(b) .

For f and [a, b] fixed, let Φε denote the function Φ from Proposition 2 corresponding

to ε , ε > 0 .

Proposition 3 With the notation above, Φε converge in C[a, b] , as ε → 0 , to the convex

hull F of the function f on the segment [a, b] .

Remark 1 In the case a ≥ b , the corresponding limit is the concave hull of f on [b, a] .

The following two assertions will be repeatedly used in the proofs in Sections 2,3:

Lemma 2 (Maximum Principle) Let Φ,Ψ ∈ C2(a, b) ∩ C[a, b] and satisfy, for all u ∈

(a, b) , the equations Φ(u) = G(u,Φ(u), Φ(u)) and Ψ(u) = H(u,Ψ(u), Ψ(u)) , respectively,

with G,H : (a, b)× IR × (0,+∞) 7→ (0,+∞] .

a) Assume that G(u, z, w) < H(u, ζ, w) for all u ∈ (a, b) such that Φ(u) < Ψ(u)

and all z, ζ, w such that z < ζ . Then Φ ≥ Ψ on [a, b] whenever Φ(a) ≥ Ψ(a) and

Φ(b) ≥ Ψ(b) .

b) Assume that G(u, z, w) ≡ H(u, z, w) , increases in w and strictly increases in z ; let

Φ(a) = Ψ(a) or Φ(b) = Ψ(b) . Then (Φ−Ψ) is monotone on [a, b] .

1.II.2. Problem (12),(13) with fixed domain 51

Proof: The proof is straightforward. ⋄

Lemma 3 Let functions F, Fn , n ∈ IN , be continuous and convex (or concave) on [a, b] .

Assume that Fn(u) converge to F (u) for all u ∈ [a, b] . Then this convergence is uniform

on all [c, d] ⊂ (a, b) and

a) Fn converge to F a.e. on [a.b] ;

b) if Fn, F are increasing, then

∫ b

a

√Fn(u)du converge to

∫ b

a

√F (u)du ;

c) let[F]−1

,[Fn

]−1

denote the graph inverse functions of F, Fn respectively; then[Fn

]−1

(ξ) tends to[F]−1

(ξ) for all ξ such that[F]−1

is continuous at the point ξ .

Proof: An elementary proof of a),c) is given in [BA1]. Besides, the assumptions of the

Lemma imply that for all δ > 0 Fn are bounded uniformly in n ∈ IN for u ∈ [a+ δ, b− δ] .Since, in addition,

∣∣∣∣∫ a+δ

a

√Fn(u)du+

∫ b

b−δ

√Fn(u)du

∣∣∣∣→ 0 uniformly in n ∈ IN as δ →0 , the conclusion b) follows from the Lebesgue Theorem. ⋄

Proof of Proposition 2: There is nothing to prove if a = b ; let a 0 (17)

for all u ∈ [a, b] . Since Gn is continuous in all variables and bounded, the existence of

solution follows for arbitrary boundary data such that Φ(a) < Φ(b) ; in particular, a solution

Φn exists such that Φn(a) = f(a) , Φn(b) = f(b) . The Maximum Principle yields that Φn

decrease to some convex non-decreasing function Φ on [a, b] as n→ ∞ .

Further, there exists a solution Ψ of (12) on [a, b] with any assigned value of Ψ(a) less

than f(a) , or any assigned value of Ψ(b) less than f(b) . In fact, in the first case one

takes Ψ(u) ≡ Ψ(a) ; in the second case there exists a solution on the whole of [a, b] to

the equation (12) with the Cauchy data Ψ(b) (fixed) and Ψ(b) sufficiently large. By the

Maximum Principle Φn ≥ Ψ on [a, b] ; therefore Φ(a + 0) = f(a) and Φ(b − 0) = f(b) .

Consequently Φ is continuous on [a, b] .

Now if for all [c, d] ⊂ (a, b) there exists m0 > 0 such that f −Φ ≥ m0 on [c, d] , then

the functions Gn(u,Φn(u), Φn(u)) are bounded uniformly in n ∈ IN for u ∈ [c, d] ; indeed,

on [c, d] , by convexity, Φn are uniformly bounded and Φn converge to Φ uniformly, so

that 2εΦn

f−Φn≤ M(c, d) for all n large enough. Hence it will follow by Lemma 3a) and the

Lebesgue Theorem that Φ(u) = 2εΦ(u)f(u)−Φ(u)

for all u ∈ [c, d] , and consequently Φ ∈ C2[c, d] .

Thus the existence of solution to problem (12),(13) will be shown.

First let show that Φ(u ± 0) > 0 for all u > a . It suffices to prove that u = a ,

where u := supu ∈ [a, b] |Φ(u) = f(a)

. Note that u f(a) .

Assume u > a ; by the Lebesgue Theorem Φ = 2εΦf−Φ

in some neighbourhood of u . Since

Φ(u − 0) = 0 , by the uniqueness theorem for the Cauchy problem Φ is constant in this

neighbourhood. Therefore necessarily u = b , which is impossible.

Further, by Lemma 3a), (17), and the Fatou Lemma one has 2εΦf−Φ

∈ L1loc(a, b) . Hence

Φ ≤ f and 2εΦf−Φ

≤ Φ on (a, b) in measure sense. Now take [c, d] ∈ (a, b) and u ∈ [c, d] ;

set m := f(u) − Φ(u) ≥ 0 . Set A := Φ(a+c2

− 0) > 0 , B := Φ(d − 0) > 0 . For all

u ∈ [a+c2, u] , f(u)− Φ(u) ≤ m+ B(u− u) and Φ(u± 0) ≥ A since Φ is convex and f

increasing. Hence

B −A ≥∫ u

a+c2

Φdu ≥∫ u

a+c2

2εΦ(u)

f(u)− Φ(u)du ≥

∫ u

a+c2

2εA

m+B(u− u)du = K1 −K2 lnm,

with some positive constants K1, K2 depending only on c, d . Thus m ≥ m0(c, d) > 0 and

the proof of existence is complete.

The uniqueness is clear from the Maximum Principle for solutions of (12). ⋄

Proof of Proposition 3: Let a 0 and a barrier function Ψα such that

α/2 ≤ F − Ψα ≤ α and Ψα ≥ m(α) > 0 on [a, b] . Such a function can be constructed

through the Weierstrass Theorem.

By the Maximum Principle Φε increase as ε decrease. Therefore there exists [c, d]

inside (a, b) such that for all ε in (0, 1) Φε ≥ Ψα on [a, b] \ [c, d] . It follows thatu |Φε(u) < Ψα(u)

⊂ [c, d] and thus Φε ≤ M(α) on this set uniformly in ε . Now

for all ε less than α·m(α)2M(α)

one may apply the Maximum Principle to Φε and Ψα , hence

0 ≤ F − Φε ≤ α for all ε small enough. ⋄

3 Solutions of the problem (6),(2) and the proof of Theorem

1

Proposition 2 above implies that for all f , ε , u± fixed, for all u0 ∈ IR there exist unique

Φε+(·; u0) and Φε

−(·; u0) satisfying (12),(13); thus by Proposition 1, for an arbitrary v− in

IR and v+ obtained from (14), (U, V ) provided by (15),(16) is a self-similar solution to the

Riemann problem (6),(2). Now since not u0 but v± are given by (2), one needs to find u0

in IR such that (14) holds with these assigned values of v± .

Proposition 4 a) Assume f(±∞) = ±∞ . Then for all u±, v± ∈ IR , ε > 0 there exists a

unique u0 such that (14) holds, with Φε+ , Φε

− the (unique) solutions to (12),(13).

b) Assume f ∈ W 11 locally in IR and

∫ ±∞

0

√f(u)du = ±∞ . Then for all u±, v± ∈ IR

and ε < ε0 = ε0(u±, v+ − v−) there exists a unique u0 such that (14) holds, with the same

Φε± .

1.II.3. Problem (6),(2) and proof of Theorem 1 53

Let F±(·; u0) be, as in the Introduction, the convex (concave) hulls of f on I(u0, u±)

according to the sign of (u± − u0) . Set

∆ε±(u0) :=

∫ u+

u0

√Φε

±(u; u0)du, ∆0±(u0) :=

∫ u+

u0

√F±(u; u0)du.

It will be convenient to extend Φε±(·; u0) , F±(·; u0) to continuous functions on IR by setting

each of them constant on (−∞,minu0, u±] and [maxu0, u±,+∞) . In the lemma below

a few facts needed for the proofs of Proposition 4 and Theorem 1 are stated.

Lemma 4 With the notation above, and u0 running through IR , the following properties

hold.

a) For all u ∈ IR and ε > 0 , u0 7→ Φε±(u; u0) do not decrease; nor do u0 7→ F±(u; u0) .

b) For all u ∈ IR and ε > 0 , u0 7→ sign(u±−u0)Φε±(u; u0) do not increase; nor do

u0 7→ sign(u±−u0)F±(u; u0) .

c) For all ε > 0 the maps u0 7→ Φε±(·; u0) are continuous for the L∞(IR) topology; so

do u0 7→ F±(·; u0) .d) For all ε ≥ 0 , u0 7→ ∆ε

±(u0) are continuous and strictly decreasing.

Proof: Combining the continuity and monotony of f with a),b) of the Maximum Principle

for solutions of (12),(13), one gets a)-c) for Φε± . The same assertions for F± follow now

from Proposition 3 and Lemma 3a); they can also be easily derived from the definition of

convex hull. Finally, d) results from c), Lemma 3b), b) and the strict monotony of f . ⋄Proof of Proposition 4: a) By Lemma 4d), it suffices to prove that ∆ε

±(±∞) = ∓∞ .

Assume the contrary, for instance that ∆ε+(−∞) =M < +∞ .

Consider u0 < u+ ; Φε+ is convex, therefore for all u0 there exists c = c(u0) ∈ [u0, u+]

such that Φε+(·; u0) ≥ 1 on [c, u+) and Φε

+(·; u0) ≤ 1 on (u0, c] . By Lemma 4b) c(u0)

increase with u0 . Obviously, for all u0 , M > ∆ε+(u0) ≥ [Φε

+(c; u0) − f(u0)] + [u+ − c] .

Set d := u+ −M ; clearly, c(u0) ≥ d for all u0 . Considering the functions Φε(·; u0) with

u0 → −∞ , one obtains a sequence Ψn such that Ψn satisfy (12) on [d, u+) , Ψn(d) ≤1 , Ψn(u+) = f(u+) , and finally, Ψn(d) → −∞ (this last since Ψn(d) ≤ f(u0) +M →f(−∞) +M = −∞ , as u0 → −∞ ). On the other hand, for n large enough, the unique

solution Ψ to the equation (12) with the Cauchy data Ψ(d) = Ψn(d) , Ψ(d) = 2 is defined

on the whole of [d, u+] , which means that Ψ(u+) < f(u+) . Now by b) of the Maximum

Principle, (Ψ − Ψn) is increasing and thus positive. Hence Ψn(u+) ≤ Ψ(u+) < f(u+) ,

which is a contradiction.

b) Take u0 < u+ . First suppose f ∈ C2[u0, u+] and has a finite number of points

of inflexion; denote by F the corresponding convex hull. The segment [u0, u+] can be

decomposed into the three disjoint sets: M1 :=u | ∃δ > 0 s.t. F ≡ const on (u − δ, u +

δ) ∩ [a, b], M2 :=

u | F (u) = f(u)

\M1 , and M3 finite. Using the Cauchy-Schwarz

inequality on every (c, d) ⊂M1 , one gets

∫ u+

u0

√F (u)du ≡ ∆0

+(u0) ≥∫ u+

u0

√f(u)du .

In the general case, let proceed with the density argument, choosing a sequence fnsuch that fn are increasing and smooth as above, fn → f in C[u0, u+] with

√fn →

√f

in L1[u0, u+] as n → ∞ . Denote the convex hull of fn on [u0, u+] by Fn ; it is easy

to see that ‖Fn − F‖C[u0,u+] ≤ ‖fn − f‖C[u0,u+] → 0 as n → ∞ . By Lemma 4b),

∆0+(u0) = lim

n→∞

∫ u+

u0

√Fn(u)du , so that ∆0

+(u0) ≥∫ u+

u0

√f(u)du in the general case as

well. Thus ∆0+(−∞) = +∞ by the assumption on f .

Now Proposition 3 and Lemma 3b) imply that for given v± in IR , there exists ε0 =

ε0(v+ − v−) such that one has ∆ε+(−L) > |v− − v+| (and in the same way, ∆ε

+(L) <

−|v−− v+| ) for all ε < ε0 whenever L is large enough. Lemma 4d) yields now the required

fact. ⋄Finally, here is the proof of the result announced in the Introduction.

Proof of Theorem 1: The existence and uniqueness of a bounded self-similar distribution

solution to the Riemann problem (6),(2) follow immediately from Propositions 1, 2 and 4.

Now let ε decrease to 0 . Take(uε0,Φ

ε±(·; uε0)

)corresponding to the unique solution

of (6),(2) in the sense of Proposition 1. Take u0 a limit point in IR of uε0ε>0 . Suppose

first uεk0 → u0 ∈ IR , εk → 0 as k → ∞ ; let show that, with the notation as in Lemma

4, Φε+(·; uε0) converge to F+(·; u0) in L∞(IR) . Indeed, take α > 0 ; |uεk0 − u0| < α for

all k large enough. By Proposition 3 and Lemma 4a), there exists ε0 > 0 such that, for all

εk < ε0 , F+(·; u0−α)−α ≤ Φεk+ (·; u0−α) ≤ Φεk

+ (·; uεk0 ) ≤ Φεk+ (·; u0+α) ≤ F+(·; u0+α)+α .

Thus the required result follows from Lemma 4c); clearly, it also holds for Φεk− , F− in place

of Φεk+ , F+ .

Now by Lemma 3b) ∆0+(u0) + ∆0

−(u0) is the limit of ∆εk+ (uεk0 ) + ∆εk

− (uεk0 ) ≡ v− − v+ ;

hence by Lemma 4d), u0 is unique if it is finite. Besides if, for instance, u0 = −∞ , then

for all L ∈ IR , v− − v+ = limεk→0[∆εk+ (uεk0 ) + ∆εk

− (uεk0 )] ≥ ∆0+(L) + ∆0

−(L) by Lemma

4d) and Lemma 3b). It is a contradiction; indeed, it is easy to see that ∆0±(L) → +∞ as

L→ −∞ .

Thus in fact uε0 → u0 as ε → 0 , u0 ∈ IR and (5) holds. Further, let u0 < u± ; the

other cases are similar and those of u0 = u− or u0 = u+ are trivial. For all α > 0 there

exists ε0 = ε0(α) > 0 such that for all ε < ε0 [uε0, u±] ⊂ [u0 − α, u±] . The functions Uε

in the statement of Theorem 1 are given by formula (15), when applied to Φε±(·; u0) with

their natural domains [uε0, u±] . Taking for the domains [u0 − α, u±] , one do not change

Uε(ξ) for ξ 6= 0 and ε < ε0 . The same being valid for U given by (3), one may use the

fact, proved above, that ‖Φε±(·; uε0) − F±(·; u0)‖C[u0−α,u±] → 0 as ε → 0 , and conclude

by Lemma 3c) that Uε(ξ) → U(ξ) for a.a. ξ ∈ IR . Hence it follows by (4),(16) that

V ε → V a.e., so that (U, V ) given by (3)-(5) is the unique a.e.-limit of self-similar bounded

distribution solutions of the problem (6),(2). Thus (U, V ) is a distribution solution of the

Riemann problem (1),(2). ⋄

1.II.4. Comments 55

Remark 2 Note that using b) of Proposition 4 instead of a), one gets a result similar to the

Theorem 1 in the case of f ∈ W 11 locally in IR ,

∫ ±∞

0

√f(u)du = ±∞ ; in fact, the exact

condition is the bijectivity of the functions u0 7→ ∆0±(u0) for continuous strictly increasing

flux function f . Under each of this conditions the existence of bounded self-similar solution

of (6),(2) is guaranteed for all ε < ε0 = ε0(u±, v+ − v−) .

Note

After this paper had been completed, the author had an opportunity to meet Prof. A.E.Tzavaras

and get acquanted with his papers on viscosity limits for the Riemann problem; in particular,

in [Tz95] very close results were obtained for p-systems regularized by viscosity terms of the

form

(0

εt(k(U)Vx)x

), without involving the explicit formulae for the limiting solution.

For results on self-similar viscous limits for general strictly hyperbolic systems of conserva-

tion laws, refer to the survey paper [Tz98] and literature cited therein. Let only note that the

structure of wave fans in self-similar viscous limits remains the same as in the case of scalar

conservation laws ([G59],[K70b]) and in the case of p-systems, where it can be easily observed

through the formulae (3),(4).

On the other hand, Prof. B.Piccoli turned my attention to Riemann solvers for hyperbolic-

elliptic systems (1) (i.e. the case of non-monotone f ). The global explicit Riemann solver

extends to this case (see Krejcı,Straskraba, [KrSt97],[KrSt93]); it can be proved, with the

techniques used here and in [BA1],[BA2], that this solver is the unique limit of self-similar

bounded solutions to the problem (6),(2).

Precise results on hyperbolic-elliptic p-systems and a discussion of other viscosity terms

are given in Section 5 below.

4 Comments

It is possible to treat, in almost the same way, the case of (by no means physical) viscosity

term

(0

εtUxx

). Lemma 1 still holds; moreover, in this case U, V ∈ C1(IR) . For Φε

±(·; u0)

defined by (11), one gets instead of (12) the equation

Φ(u) =2ε√Φ(u)

f(u)− Φ(u)with Φ > 0 and Φ > 0 or Φ < 0;

this problem shares the properties of (12) that were important for us.

Further, the more general viscosity terms

(0

εt(k(U)Vx)x

)or

(0

εt(k(U)Ux)x

), with

k ∈ C(IR; (0,+∞)) , also yield unique approximate solutions, which converge to (U, V ) given

by (3)-(5). The only difference is the factor k(u) in the right-hand side of the equation in

(12).

Finally, the classical example of p-system with f(u) = 1/u , k(u) = 1/u on the domain

u > 0 , given by the isentropic gas dynamics in Lagrangian coordinates, can be included; as

in b) of Proposition 4, the divergence of the integral

∫ √f(u)du at 0 and +∞ guarantees

existence of solution to (6),(2) for all data u± > 0 , v± ∈ IR .

On the contrary, in the case of γ -pressure laws with γ > 1 (i.e. f(u) = 1/uγ ) we

have to impose the restriction v+ − v− <

∫ +∞

u−

√f(u)du +

∫ +∞

u+

√f(u)du in order to

have existence of a bounded self-similar solution to the approximating system. Otherwise, we

have to deal with solutions where u is unbounded, which corresponds to vanishing density

ρ . Still the vaccuum remains invisible in Lagrangian representation†, so that the appropriate

description of such problems has to be given in the Eulerian one. In Chapter 1.III, we will

carry out a thoroughful investigation of the Riemann problem for the system of isentropic

gas dynamics in Eulerian coordinates. Similar results will be obtained for general continuous

pressure laws and arbitrary Riemann data, including those that give rise to vaccuum in the

solutions.

Another extension of techniques applied above is presented in Section 5 below, where we

treat the Riemann problem for hyperbolic-elliptic systems of form (1). Indeed, as it is shown in

†Unless one considers measure-valued solutions. In [Wa87] the equivalence of equations of gas dynamics

in Eulerian and Lagrangian coordinates is shown for a very large class of weak solutions. Moreover, the

transformation between the two representations preserves the class of convex entropies, so that it preserves

the entropy admissibility in the sense of Lax (cf. [K70a, Lax71]).

In particular, the result of [Wa87] applies to the Riemann problem with the two initial states different from

vaccuum. It would be interesting to check whether Wagner’s arguments apply to viscosity regularized systems

(which is, clearly, the case when there is no vaccuum). If yes, the wave-fan admissibility is preserved under the

change of representation. As show the results of Chapter 1.III, the solutions to the Riemann problem (6),(2) are

still ordinary functions, though may be unbounded when the assumption

∫ +∞

1

√f(u)du+

∫ +∞

1

√f(u)du =

+∞ fails. But we have to pass to the limit in the sense of Radon measures, since the Lagrangian equivalent

of the solution (40)-(45) in Chapter 1.III can contain a Dirac mass (a so-called δ -wave).

There is a connection with another interesting question. Namely, one system where δ -waves naturally arise

in solutions to the Riemann problem is the nonstrictly hyperbolic system

ut + (u2/2)x = 0

vt + (uv)x = 0(∗)

considered by K.T.Joseph in [Jo93]. This is the simplest representative of another class of systems where the

idea of the present work could be applied for solving the Riemann problem. Complications will arise of the

same order as indicated above. Still it indicates that there could be an explicit formula for admissible solutions

of the Riemann problem. Not surprisingly, this formula does exist for the system (∗) : it is obtained in [Jo93],

through the viscous regularisation

(εuxx

εvxx

)and the Hopf-Cole-Lax transformation (cf. [H50, Lax57]).

1.II.5. Hyperbolic-elliptic case 57

[KrSt93] (cf. [KrSt97] for a detailed exposition), the monotony of f is not essential; under the

assumption that there exist (finite or infinite) limits f± =: limu→±∞

f(u) and f− < f(u) < f+

for all u , one can justify the appropriate version of formulae (3)-(5) through the maximal

dissipation principle of Krejcı-Straskraba. We give a justification of this formulae by the self-

similar vanishing viscosity approach.

5 The hyperbolic-elliptic case

Let consider the problems (1),(2) and (6),(2) without the monotony assumption on f . In-

stead, we require that

f− < f(u) < f+ for all u, where f− =: lim infu→−∞

f(u), f+ =: lim supu→+∞

f(u). (18)

Self-similar viscosity limits for a class of hyperbolic-elliptic p-systems modelling van der Waals

fluids have been constructed in [Sl89] and [Fan92], in case of the identity viscosity matrix. It

generates solutions that have the same structural properties as the solution (19)-(22) obtained

below as the limit of solutions of (6),(2). In particular, only stationary phase transitions (i.e.,

jumps across elliptic regions) occur in the two cases. This is not always satisfactory; some

comments on this issue are presented in Remark 4 at the end of this section.

As before, let restrict our attention to bounded self-similar solutions. Let (U, V ) be such

a solution to (6),(2) with some ε > 0 . In general, U will not be continuous at ξ = 0 , since

the continuity of f(U) at 0 yields it no more. Neither will U be strictly monotone outside

of a neighbourhood of ξ = 0 . In turn, this implies that the functions Φ±(·) , defined as in

(11), do not necessarily verify (12) in the classical sense. Whence the two modifications we

have to perform, comparing to the case of strictly increasing f .

First, we take f0 = f(U)(0) , which is well defined, for the parameter in our study

of viscous solutions corresponding to given u± and v− . It will be shown that the limits

u0,± := limξ→±0

U(ξ) and the value of v− − v+ are uniquely defined by f0 . The relation with

the problem (12)-(13) on the intervals I(u0,±, u±) , now understood in a weakened sense

similar to (5) in Chapter 1.I, will be established, followed by the existence and uniqueness result

for this problem. Second, in order to overcome the loss of regularity relative to possible sharper

singularities in f , we will follow the ideas carried out in the case of scalar equation (Chapter

1.I). Beyond some minor modifications in other arguments, we have to upgrade Lemma 1 (the

monotony part), Proposition 1 and the Maximum Principle; in turn, this simplifies the proof

of the version of Proposition 2.

Let f0 ∈ (f−, f+) . Under the assumption (18), there exist (unique) u0,± that satisfy

u0,± :=

maxu < u± | f(u) = f0, f0 < f(u±)

u±, f0 = f(u±)

minu > u± | f(u) = f0, f0 > f(u±).

(19)


Denote by F+ = F+(·; u0,+) the convex hull of f on [u0,+, u+] if u0,+ ≤ u+ , and the

concave hull of f on [u+, u0,+] if u0,+ ≥ u+ . Replacing u+ by u− and u0,+ by u0,− ,

define F− = F−(·; u0,−) in the same way. We will prove that the unique viscous limit solution

to (1),(2) is given by the formulae

U(t, x) =

[F+(·; u0,+)

]−1

(x2/t2), x ≥ 0[F−(·; u0,−)

]−1

(x2/t2), x ≤ 0, (20)

V (t, x) = v− −∫ x/t

−∞

ζdU(ζ), (21)

dU(ζ) being regarded as measure; (19) and the formula

v− − v+ =

∫ u+

u0,+

√F+(u; u0,+)du+

∫ u−

u0,−

√F−(u; u0,−)du. (22)

provide a one-to-one correspondence between f0 ∈ (f−, f+) and an interval of admissible

values of v− − v+ . Additional conditions are required in order to have all values of v− − v+

admissible; for example, f± = ±∞ is sufficient (see [KrSt93]).

Let start with

Lemma 5 A pair of bounded functions (U, V ) : ξ ∈ IR 7→ IR2 is a self-similar distribution

solution of (6),(2) if and only if:

· V, V ′, ξU ′ are continuous on IR , U is continuous on IR\0 and admits limits u0,±

as ξ → ±0 such that f(u0,−) = f(u0,+) ;

· the equations

εξU ′(ξ) = −∫ ξ

0

ζ2U ′(ζ)dζ + f(U(ξ)) + C, (23)

V (ξ) = −∫ ξ

0

ζU ′(ζ)dζ +K (24)

are fulfilled with some constants C,K ;

· U(±∞) = u±, V (±∞) = v±. (25)

Besides, U, V are monotone on each of (−∞, 0) , (0,+∞) .

In addition, u0,± are connected to f0 := f(u0,−) = f(u0,+) through (19).

Proof: Just as in the proof of Lemma 1, we first deduce (23) for ξ > 0 and ξ < 0 .

We would arrive to U ∈ C(IR \ 0) , ξU ′(ξ) ∈ C(IR) with limξ→±0 ξU′(ξ) = 0 , f(U) ∈

C(IR) , and V ∈ C1(IR) , if we only could show that there exist u0,± = limξ→±0U(ξ) .

Again, it follows from the monotony of U on both sides from the singularity point ξ = 0 .

The monotony property can be proved using (23) and the reasoning carried out in Lemma 1

of Chapter 1.I (see also [BA1]). We complete the proof of the equivalence as in Lemma 1.

Further, note that C = −f0 . For instance, let f(u−) < f0 . The formula (23) together

with the monotony of U imply that, first, sign (u−−u0,−) = sign (f(u−)− f(u0,−)) = −1 ;

second, for all u ∈ [u−, u0,−) , f(u) < f0 . These two conditions yield u0,− = minu >

u− | f(u) = f0 . Thus (19) holds for this case; the other cases are similar. ⋄

Let (U, V ) be a bounded self-similar solution of (6),(2). Using the monotony of U on

(−∞, 0) and (0,+∞) , one can inverse it on each of these intervals. The resulting functions

U−1± are defined a.e. on I(u0,±, u±)

† and monotone. For all u ∈ I(u0,±, u±) , set

Φε±(u; u0,±) :=

∫ u

u0,±

(U−1± (w)

)2dw − C (26)

with C taken from (23). We will abrige Φε±(·; u0,±) to Φ±(·) whenever ε and u0,± are

fixed. Now (23) can be rewritten as

εξU ′(ξ) = f(U(ξ))− Φ±(U(ξ)) for ξ ∈ I(0,±∞). (27)

As in Chapter 1.I, set Ω± := u | Φ±(u) = f(u) ≡ u | ∃ξ ∈ IR± such that U(ξ) =

u, U ′(ξ) = 0 . By the Sard lemma, Ω± have the Lebesgue measure 0 . We still obtain

that the equation in (12) is fulfilled outside of Ω± in classical sense. But on the whole of

I(a, b) = I(u0,±, u±) , we only have (12),(13) fulfilled in the following sense:

Φ ∈ C(I(a, b)), Φ is strictly increasing and (b− a)Φ is convex on I(a, b);

(b− a)(f − Φ) ≥ 0 on I(a, b) and Φ(a) = f(a),Φ(b) = f(b);

(b− a)G(·,Φ(·), Φ(·)) ∈ L1loc(I(a, b)) ∩ C(I(a, b); (0,+∞]);

(b− a)(Φ(·)−G(·,Φ(·), Φ(·))

)≥ 0 in the measure sense on I(a, b);

(f(·)− Φ(·))(Φ(·)−G(·,Φ(·), Φ(·))

)= 0 in the measure sense on I(a, b)

(28)

with G : (u, z, w) ∈ I(a, b) × (−∞,+∞) × (0,+∞) 7→ 2εwmaxf(u)−z,0

∈ (0,+∞] . In-

deed, for instance suppose u0,− < u− . Then U−1− is a non-increasing negative function.

Hence Φ is strictly increasing and Φ−(u) = ddu

(U−1−

)2is a non-negative measure on

(u0,−, u−) . By (27), Φ− ≤ f and Φ− can be extended on I(a, b) by continuity, with

Φ−(a) = f(a),Φ−(b) = f(b) . Since 2εΦ−

f−Φ−= Φ− on (u0,−, u−)\Ω− and Ω− is of measure

0 , 2εΦ−

f−Φ−∈ L1

loc(u0,−, u−) . The last two properties in (28) are now evident. Besides, it follows

†As above, for a, b ∈ IR , we denote by I(a, b) the interval (mina, b,maxa, b)

from (19) that f(u0,−) < f(u) for all u ∈ (u0,−, u−] ; this implies, as in the proof of Propo-

sition 2, that Φ− > 0 on (u0,−, u−] . Thus we can write 2εΦ−

f−Φ−∈ C((u0,−, u−]; (0,+∞]) .

Conversely, a pair of functions Φ± verifying (28) on I(u0,±, u−) , where u0,± satisfy

(19), gives rise to a bounded self-similar solution of (6), (2) through the formulae

U(ξ) =

[Φ+(·; u0,+)

]−1

(ξ2), ξ > 0[Φ−(·; u0,−)

]−1

(ξ2), ξ < 0,(29)

V (ξ) = v− −∫ ξ

−∞

ζdU(ζ) (30)

under the following condition that generalizes (14):

v− − v+ =

∫ u+

u0,+

√Φε

+(u; u0,+)du+

∫ u−

u0,−

√Φε

−(u; u0,−)du. (31)

Indeed, (25) is obvious. As above, (28) implies that Φ− > 0 on (u0,−, u−] ; further, as in

Lemma 2 in Chapter 1.I, we deduce (27) for all ξ ∈ IR . Hence (29) yields (23), (24) and the

continuity properties of Lemma 5.

We summarize Lemma 5 and the results above in

Proposition 5 Let ε, f, u±, v± be fixed. Formulae (29),(30) provide a one-to-one corre-

spondence between the sets A and B defined by

A :=(f0,Φ±(·)

)| f0 ∈ IR, Φ± : I(u0,±, u±) 7→ IR and (28), (31) hold,

where u0,± are given by (19)

B :=(U, V ) | (U, V ) is a bounded self-similar distribution solution of (6), (2)

Remark 3 Note that (28) is equivalent to (12),(13) under additional regularity assumptions

on f . For example, it is sufficient that f be sublinear at least on one side from each point

of IR . The argument is the same that was used in the proof of Proposition 2.

We state the existence and convergence results and the maximum principle for (28) in case

a 0 and a, b ∈ IR such that f(u) >

f(a) for all u ∈ (a, b] , there exists a unique Φ that satisfy (28).

Proof: Using the Maximum Principle below, as in the proof of Proposition 2 we construct a

sequence of solutions Φn to (12) penalized by troncatures. (We need the condition f > f(a)

on (a, b] in order to guarantee Φn > 0 .) As n → +∞ , Φn decrease to a continuous

increasing convex function Φ on [a, b] .

By the Fatou Lemma, Φ ≥ 2εΦf−Φ

in the sense of measures on (a, b) ; this also implies that

Ω := u | Φ(u) = f(u) is of measure 0 , and Φ ≤ f . Besides, if Φ(u) < f(u) , then1

f−Φnand Φn are uniformly bounded in a neighbourhood of u , because Φn are convex

and Φn(b) = Φ(b) < Φ(a) for all n . Therefore we get the equality in (28) outside of

Ω . As in Proposition 2, it follows that Φ is strictly increasing. Other properties in (28)

are now obvious. Thus the existence of solution is shown; the Maximum Principle yields the

uniqueness. ⋄

Lemma 6 (Maximum Principle) Let (a, b) ⊂ IR and Φ,Ψ satisfy, in the sense of (28),

the “inequalities” Φ(u) ≥ G(u,Φ(u), Φ(u)) and Ψ(u) ≥ H(u,Ψ(u), Ψ(u)) , respectively,

with G,H : (a, b)× (−∞,+∞)× (0,+∞) 7→ (0,+∞] .

a) Assume that G(u, z, w) < H(u, ζ, w) for all u such that Φ(u) < Ψ(u) and all z, ζ, w

such that z < ζ . Then Φ ≥ Ψ on [a, b] whenever Φ(a) ≥ Ψ(a) and Φ(b) ≥ Ψ(b) .

b) Assume that G(u, z, w) ≡ H(u, z, w) and increases in z (strictly) and in w ; let

Φ(a) = Ψ(a) or Φ(b) = Ψ(b) . Then (Φ−Ψ) is monotone on [a, b] .

Proof: For instance, let prove a). Take c the point of minimum of Φ − Ψ ; c ∈ (a, b) .

Suppose this minimum is negative. Then Φ(c) < Ψ(c) ≤ f(c) , so that there exists δ1 > 0

such that Φ(·) = G(·,Φ(·), Φ(·)) ∈ L1loc(c, c+ δ1) . Then there exists Φ(c) ; by the choice of

c , we will have Ψ(c+0) ≤ Φ(c) ≤ Ψ(c−0) . The convexity of Ψ implies that Ψ(c) = Φ(c) .

Therefore we have G(u,Φ(u), Φ(u)) < H(u,Ψ(u), Ψ(u)) for u = c , and consequently, for

all u ∈ [c, c + δ2) for some δ2 > 0 . From (28) we have Φ < Ψ in the sense of measures

on [c, c + δ) , where δ := minδ1, δ2 . Therefore Φ − Ψ strictly decreases on [c, c + δ] ,

which contradicts to the choice of c . ⋄Other changes, with respect to the case of monotone f , are minimal. Let only note that

we have to take into account that u0,± are not continuous as functions of f0 , in particular

in the assertion analogous to d) of Lemma 4.

As in the continuous case, we can prove the convergence of Φε± with fixed f0 ; then

establish that (31) is a bijection between f0 and v−−v+ (for instance, under the assumption

f± = ±∞ ); then prove the convergence of Φε± with fixed v− − v+ to the corresponding

convex hulls and apply Lemma 3.

The following theorem holds:

Theorem 2 Suppose lim infu→−∞

f(u) = −∞, lim supu→+∞

f(u) = +∞ . Then for all u±, v± ∈ IR ,

ε > 0 there exists a unique bounded self-similar distribution solution (Uε, V ε) of the problem

(6),(2).


Besides, as ε ↓ 0 (Uε, V ε)(ξ) → (U, V )(ξ) a.e. on IR , where (U, V ) is given by

the formulae (19)-(22), so that (U, V ) is a self-similar distribution solution of the problem

(1),(2).

Remark 4 In this last section we have answered to a question posed by R.James in [Ja80],

the paper from which an extensive study of admissibility for (1),(2) in the non-monotone case

had started. James studies nonlinear elasticity and proposes to define admissible solutions as

limits of the vanishing viscosity approximations of the formUt − Vx = 0

Vt − f(U)x = εVxx,(32)

following the idea defended by Rayleigh ([Ray10]) for the isentropic fluid dynamics. James

calls this criterion the viscoelastic criterion. He proceeds by deducing that the energy should

decrease in time, but this condition is not sufficient for uniqueness for the Riemann problem

(which is well known in the scalar case, in absence of convexity). He proposes then to

obtain more restrictions on admissible shocks either by using the full strength of the viscosity(0

εVxx

), either by replacing this viscosity in (32) by the artificial one used by Dafermos in

[D74],

(εtUxx

εtVxx

)(or by

(0

εtVxx

), which encounters less physical objections).

For the first case, there is a response in the case of flux functions with one inflexion point.

M.Shearer in [Sh82] constructs a Riemann solver which provides a unique solution to (1),(2) for

all data, using classical shocks (always admissible by the viscoelastic criterion) and additional

phase transitions propagating with zero speed. R.Pego in [Pe87] observes that only these two

kinds of shocks can be approximated by travelling waves solutions of (32).

The result of this section give a response for the second case. Evidently, in the Riemann

solver (19)-(22) the only admissible transitions between elliptic and hyperbolic regions or

between two elliptic regions are those with zero speed.

It seems that this last property is not always what one observes in physical systems. The

reason is, dissipation effects captured by vanishing viscosity can coexist with dispersion effects

provoked by capillarity. M.Slemrod in [Sl83] proposes a family of viscosity-capillarity criteria,

regularizing (1) with

(0

εVxx −Aε2Vxxx

). The parameter A ∈ [0, 1/4] regulates phase

transitions; an important feature is that viscosity-capillarity limits admit phase transitions of

non-zero speed whenever A > 0 . Clearly, the self-similar viscosity approach exposed above

is unable to capture this kind of effects. To pursue the study, one has to introduce additional

self-similar dissipation, as it has been done by M.Slemrod and H.Fan in [Sl89, Fan92].

Observations on the difference of diffusive and diffusive-dispersive limits for (1),(2) have

recently gave rise to a theory of non-classical Riemann solvers, in particular for nonconvex scalar

conservation laws. A survey of results on diffusive-dispersive limits and related questions can

be found in [LF98].

CHAPTER 1.III

On Viscous Limit Solutions

to the Riemann Problem

for the Equations of Isentropic Gas Dynamics

in Eulerian Coordinates†

Introduction

In this chapter we study bounded self-similar solutions to the problem

ρt + (ρu)x = 0

(ρu)t + (ρu2 + p(ρ))x = εtuxx(1)

with the initial condition

ρ(0, x) =

ρ+, x > 0

ρ−, x < 0, u(0, x) =

u+, x > 0

u−, x < 0(2)

and establish convergence of solutions as ε ↓ 0 .

Within the framework of isentropic gas dynamics in Eulerian coordinates, (ρ, u) : (t, x) ∈IR+ × IR 7→ (ρ(t, x), u(t, x)) ∈ IR+ × IR corresponds to the density and velocity in gas,

p is the pressure law of the gas, and ε > 0 models small dissipation of the momentum.

We assume that ρ± > 0 and p(·) is continuous strictly increasing on IR+ , normalized by

p(0) = 0 .

Recently, the problem (1),(2) has been treated in [Kim99], following the ideas of [Tz95] (see

also [Tz96]). Under additional assumptions that prohibit vaccuum in the solutions, existence

for the problem (1),(2) was proved. The set of all solutions was shown to be compact in BV ,

and the wave-fan structure of limiting functions as ε ↓ 0 was described.

†This chapter is being prepared upon publication [BA4]

64 Riemann Problem for Gas Dynamics in Eulerian Coordinates

Here we give a description of solutions to (1),(2) that is also valid when a solution contains

a vaccuum state. It also suggests a formula for the limiting function as ε ↓ 0 (cf. Section 3).

Let illustrate the problem with the classical example of γ -laws, i.e. p(ρ) = const · ργ ,γ ≥ 0 . First take γ = 1 . Set f(V ) = −p(1/V ) , k(V ) = 1/V , and consider the problem

Vt − uy = 0

ut − f(V )y = εt(k(V )uy)y(3)

V (0, y) =

1/ρ+, y > 0

1/ρ−, y < 0, u(0, y) =

u+, y > 0

u−, y < 0. (4)

This problem is (1),(2) rewritten in the Lagrangian coordinates (here y is the matherial

coordinate, and V = 1/ρ ), provided ρ > 0 (e.g., see [RoJa], [ChHs]). For the system

(3), the Riemann problem can be studied extensively (cf. Section 4 of Chapter 1.II) by the

method applied in [BA3]. It yields existence and uniqueness of a bounded self-similar solution

for all Riemann data V± > 0, u± ∈ IR , and all ε > 0 . As ε ↓ 0 , the solutions converge

to a function described by an explicit formula, based on use of convex and concave hulls of

the graph of f . In each of this solutions V is bounded, i.e. ρ = 1/V > 0 .Therefore we

can pass to the Eulerian coordinates and deduce the same results for (1),(2). The explicit

formula for the limiting function will use the images of convex/concave hulls of f(V ) under

the transformation

T : [F : V ∈ (0,+∞) 7→ F (V )] 7→ [P : ρ ∈ (0,+∞) 7→ −F (1/ρ)]. (5)

For instance, let [a, b] ⊂ (0,+∞) and F (·) be the concave hull of f(·) on [1/b, 1/a] . Let

P (·) be the function ρ ∈ (0,+∞) 7→ −F (1/ρ) . Then P (·) can be characterized by the

following properties:

(i) P (·) ∈ C[a, b] and P ≤ p on [a, b]

(ii) the function F = T−1P is concave on (1/b, 1/a)

(iii) for all Q(·) that satisfies (i) and (ii), one has P ≥ Q on [a, b].

(6)

In the case p(ρ) = const · ργ with γ > 1 , it is well known that, in general, one cannot

avoid the appearance of vaccuum in solutions of (1),(2) with ε = 0 (a detailed study of

this problem for γ ≥ 1 through construction of wave curves on the half-plane (ρ, u) can be

found in [ChHs]). The same difficulty appears for ε > 0 . This is due to the fact that the

integral

∫ 1

ρ

√d

drp(r)

dr

r=

∫ 1/ρ

1

√d

dvf(v)dv converges as ρ → 0 , which impose a bound

on the size of Riemann data in order to have a bounded solution of (3),(4) (e.g., see Chapter

1.II and [BA3]). We will solve the problem (1),(2) independently (cf. Theorem 1) since we

cannot reduce it to (3),(4) any more†.

†In fact, such reduction is possible, but should involve measure-valued solutions at the limit as ε ↓ 0 ; see

[Wa87] and the footnote in Section 4 of Chapter 1.II.

1.III.0. Introduction 65

Nevertheless, it is still possible to deduce a formula for the limiting function. In fact, (6)

makes sense also for a = 0 (upon formally setting 1/a = +∞ in (ii)), which corresponds

to the presence of a vaccuum state ρ = 0 . More precisely, for all [a, b] ⊂ [0,+∞) there

exists a (unique) function P (·) that satisfies (6). Indeed, for a > 0 it is obvious; for a = 0 ,

we construct P (·) as the decreasing limit of functions Pδ(·) such that Pδ(·)|[0,δ] ≡ p(δ) ,

Pδ(·)|[δ,b] verifies (6) on [δ, b] . Proprieties (i)− (iii) in (6) follow easily from the monotony

and continuity of P (·) . This motivates the following definition.

Definition 1 Let p : IR+ 7→ IR be continuous and strictly increasing. For a ≥ 0 and

b ≥ a , the lower (-1)-hull of p(·) on [a, b] is the function P (·) that verifies (6). For b > 0

and a ≥ b , the upper (-1)-hull of p(·) on [b, a] is the function P (·) that verifies

(i) P (·) ∈ C[b, a] and P ≥ p on [b, a]

(ii) the function F = T−1P is convex on (1/a, 1/b)

(iii) for all Q(·) that satisfies (i) and (ii), one has P ≤ Q on [b, a].

(7)

For a′, b′ ∈ (IR+)2 denote the segment [mina′, b′,maxa′, b′] by I(a′, b′) . The (-1)-hull

of p(·) on I(a′, b′) is the lower (-1)-hull on [a′, b′] in case a′ ≤ b′ and the upper (-1)-hull

on [b′, a′] in case a′ > b′ .

In Section 3 we consider the system

ρt + (ρu)x = 0

(ρu)t + (ρu2 + p(ρ))x = 0(8)

and, using (-1)-hulls of p(·) , give a formula for the unique solution of the Riemann problem

(8),(2) that can be obtained as a limit of self-similar bounded weak solutions of (1),(2) with

ε = εn for a sequence εn ↓ 0 .

The approach by self-similar viscosity limits has been used in [Ka59, Tu64, Tu66, Tu73,

D73a, D74, DDp76] in the context of admissibility of weak solutions to the Riemann problem

for hyperbolic systems of conservation laws. In [D89], Dafermos postulated it as the wave fan

admissibility criterion. It has been successfully tested on various special systems and viscosity

matrices; see [Tz98], [Tz96] for a survey of recent results in this direction. In particular, an

analysis of the problem (8),(2) regularized with the self-similar viscosity εt

(ρ

ρu

)is carried

out in [SlTz89], covering among others the cases where vaccuum is present. Within the same

framework, a special attention to the formation of the vaccuum state is paid in [Fan91]. In

[Tz96] the existence of an admissible solution is proved for a large class of strictly hyperbolic

systems with close Riemann data, using the identity self-similar viscosity matrix.


The main result of this chapter is that, for the case of the degenerate viscosity matrix(0 0

0 1

)and under the additional assumption

(a) either p(ρ) → +∞ as ρ→ +∞,

(b) or p ∈ W 1,1loc (R,+∞) for some R > 0, and d

∫ +∞

R

√d

drp(r)

dr

r= +∞,

(9)

there is global existence and uniqueness of an admissible weak solution to the Riemann problem

for the nonstrictly hyperbolic system (8) in the sense of the wave fan admissibility criterion

(cf. Theorem 2). We describe the structure of this solution in Section 3.

1 Some useful properties of viscous approximations

Let fix ε > 0 . We are concerned with bounded self-similar distribution solutions to the

problem (1),(2).

Definition 2 A pair of functions (ρ, u) : IR+ × IR 7→ IR+ × IR is a solution of the problem

(1),(2) if for all k > 0 , (ρ, u)(t, x) = (ρ, u)(kt, kx) for a.a. (t, x) ∈ IR+ × IR , ρ, u ∈L∞(IR+ × IR) , (1) is fulfilled in D′(IR+ × IR) , and

ess limt↓0

(‖ρ(t, ·)− ρ(0, ·)‖L1(−R,R) + ‖u(t, ·)− u(0, ·)‖L1(−R,R)

)= 0 (10)

for all R > 0 , where ρ(0, ·), u(0, ·) are given by (2).

We will denote x/t by ξ and ambiguously use the same notation for a self-similar function

of the variables (t, x) and the corresponding function of ξ .

Lemma 1 A pair (ρ, u) is a solution of (1),(2) in the sense of Definition 2 if and only if the

following conditions are fulfilled:

(i) there exist continuous bounded functions ρ, u : IR 7→ IR , with u′(·) and (·−u(·))ρ′(·)continuous, such that (ρ, u)(t, x) = (ρ, u)(x/t) for a.a. (t, x) ∈ (0,+∞)× IR ;

(ii) there exists a constant C ∈ IR such that one has

εu′(ξ) = −∫ ξ

0

(ζ − u(ζ))2ρ′(ζ) dζ + f(ρ(ξ)) + C, (11)

ρ(ξ)u′(ξ) = (ξ − u(ξ))ρ′(ξ), (12)

limξ→±∞

ρ(ξ) = ρ±, limξ→±∞

u(ξ) = u±. (13)

1.III.1. Useful properties of viscous approximations 67

Besides, ther exists a unique ξ0 such that u(ξ0) = ξ0 . In case ρ(ξ0) > 0 there exist

ξ± ∈ IR , ξ− ≤ ξ0 ≤ ξ+ , such that both ρ(·) and u(·) are constant on (ξ−, ξ+) and

strictly monotone on (−∞, ξ−) and (ξ+,+∞) .

In case ρ(ξ0) = 0 , ξ0 is the unique vaccuum point in the solution, ξ± = ξ0 , and the same

monotony properties hold.

Moreover, u′(ξ) 6= 0 for all ξ ∈ (−∞, ξ−) ∪ (ξ+,+∞) , u′(ξ0) = 0 in case ρ(ξ0) > 0 and

0 ≤ u′(ξ0) < 1 in case ρ(ξ0) = 0 .

Remark 1 We see from Lemma 1 that in case ε > 0 , solutions of (1),(2) contain at most

one vaccuum point.

Proof: The proof consists of four steps.

I) Let (ρ, u) be solution of (1),(2). Then (ρ, u)(t, x) = (ρ, u)(x/t) and we have in D′(IR)

−ξρ′ + (ρu)′ = 0

−ξ(ρu)′ + (ρu2 + p(ρ))′ = εu′′.(14)

It follows that εu′′ = −ξ2ρ′ + (ρu2 + p(ρ))′ = −(ξ2ρ)′ + 2ξρ + (ρu2 + p(ρ))′ in D′(IR) ,

whence u′ ∈ L∞loc(IR) and u ∈ C(IR) . Thus ρu′, (ρu)u′ are well defined in D′(IR) , hence

ρ′u , ρ′u2 as well. Therefore we obtain

−(ξ − u)ρ′ + ρu′ = 0 in D′(IR) (15)

and

−(ξ − u)2ρ′ + p(ρ)′ = εu′′ in D′(IR). (16)

It follows that −(ξ − u)ρ =

∫ ξ

0

ρ(ζ) dζ + const ∈ C(IR) . Consequently, −(ξ − u)2ρ +

p(ρ)− εu′ =

∫ ξ

0

2(ζ − u(ζ))ρ(ζ) dζ + const ∈ C1(IR) . Thus

p(ρ(·))− εu′(·) ∈ C(IR). (17)

II) Now consider the function ξ 7→ ξ−u(ξ) . It is continuous and tends to ±∞ as ξ → ±∞ .

Therefore there exist finite η− := minξ | u(ξ) = ξ and η+ := maxξ | u(ξ) = ξ . One

has ξ − u(ξ) < 0 on (−∞, η−) and ξ − u(ξ) > 0 on (η+,+∞) . By (15), on each of

these intervals one has ρ′ ∈ L∞loc . Therefore ρ is continuous on these intervals; by (17), so

does u′ ; by (15), so does ρ′ .

Let prove that ρ, u are monotone on (−∞, η−) and (η+,+∞) . Take ξ+ := supξ ≥η+ |ρ|(η+,ξ) ≡ const . First show that u is strictly monotone on (ξ+,+∞) . Indeed, suppose

the contrary. Then there exists c ∈ (ξ+,+∞) and 0 < δ < c − ξ+ such that u′(c) = 0

and u′ 6= 0 on (c− δ, c) . From (15) we have ρ′ = ρu′

ξ−u≥ 0 on (c− δ, c) , so that p(ρ(·))

is non-decreasing on [c− δ, c] . Hence by (15),(16) we have

εu′(ξ) =

∫ c

ξ

(ζ − u)ρu′ dζ + p(ρ(ξ))− p(ρ(c)) ≤∫ c

ξ

(ζ − u)ρu′ dζ

pointwise on [c − δ, c] . Choosing a sequence ξn ↑ c such that u′(ξn) = maxξ∈[ξn,c] u′(ξ) ,

we get εu′(ξn) ≤ u′(ξn) O(c − ξn) , where O(·) is the Landau symbol. As n → ∞ , we

deduce ε ≤ 0 , which is a contradiction.

We conclude that u is strictly monotone on (ξ+,+∞) ; hence ρ is monotone on

(η+,+∞) . Similarly, there exists ξ− ≤ η− such that u, ρ are strictly monotone on

(−∞, ξ−) and ρ ≡ const on [ξ−, η−) .

III) Let investigate the behaviour of ρ, u on the interval [η−, η+] . First note that there exist

finite limits ρ(η± ± 0) ; by (17), there exist finite limits u′(η± ± 0) . We notice that, first,

u′(η+ + 0) = 0 in case ρ(η+ + 0) > 0 , and 0 ≤ u′(η+ + 0) < 1 in case ρ(η+ + 0) = 0 .

The same relation exists between u′(η− − 0) and ρ(η− − 0) .

Indeed, let ρ(η++0) > 0 . By (15), we have

∫ η++1

η+

(ln ρ(ζ)

)′dζ =

∫ η++1

η+

u′(ζ)

ζ − u(ζ)dζ .

The integral on the left converge, therefore the limit u′(η + 0) is necessarily 0 . Further, let

ρ(η++0) = 0 . Then ρ is non-decreasing on (η+,+∞) , hence u′(η++0) ≥ 0 by (15). On

the other hand, the definition of η+ trivially implies that u′(η+ + 0) ≤ 1 . Besides, suppose

u′(η+ + 0) = 1 . From (16),(15) we obtain

ε(1− u′(ξ)) =

∫ ξ

η+

(ζ − u(ζ))ρ(ζ)u′(ζ) dζ − p(ρ(ξ))

for all ξ ∈ (η+,+∞) . Since ρu′ is continuous on this interval and tends to 0 as ξ ↓ η+ ,

there exists δ > 0 such that

ε(ξ − u(ξ))′ ≤∫ ξ

η+

(ζ − u(ζ)) dζ (18)

for all ξ ∈ (η+, η+ + δ) . Setting g(ξ) := ξ − u(ξ) and h(ξ) :=√εg(ξ) +

∫ ξ

η+

g(ζ)dζ , we

obtain from (18) that h ∈ C1[η−,+∞) and h′(ξ) ≤ 1/√ε h(ξ) . Since h(η+) = 0 , we get

h ≡ 0 on [η+, η+ + δ] by the Gronwall inequality. This contradicts the definition of η+ .

Thus, definitively, 0 ≤ u′(η+ + 0) < 1 . The proof for η− − 0 in the place of η+ + 0 is

likewise.

Now we analyse separately the three cases:

a) η− = η+ =: ξ0 and one of the limits ρ(ξ0 ± 0) is non-zero. Then the two limits

coincide. Indeed, let ρ(ξ0 + 0) > 0 . First assume ρ(ξ0 − 0) > 0 ; in this case u′(ξ0 ± 0) =

0 , so that ρ(ξ0 ± 0) coincide by (17). Further, assume ρ(ξ0 − 0) = 0 . In this case

p(ρ(ξ0+0))−εu′(ξ0+0) = p(ρ(ξ0+0)) > 0 , and p(ρ(ξ0−0))−εu′(ξ0−0) = −εu′(ξ0−0) ≤ 0 ,

which contradicts (17). Hence we see that ρ, u′ are both continuous on IR . Therefore

(15),(16) can be rewritten under the form (11),(12).

We see that in this case there exists a unique ξ0 such that u(ξ0) = ξ0 , and we have

ρ(ξ0) > 0 and u′(ξ0) = 0 . Besides, with ξ± defined above, we find that ρ, u are constant

on [ξ−, ξ+] and that u′ , consequently ρ′ , are different from 0 on (−∞, ξ−)∪ (ξ+,+∞) .

b) η− = η+ =: ξ0 and ρ(ξ0 ± 0) = 0 . Then ρ, u′ are continuous on IR , and (11),(12)

follow. Further, ξ = ξ0 is the unique vaccum point. Indeed, for instance, take ξ+ =

supξ | ρ(ξ) = 0 . We have

∫ ξ++1

ξ+

(ln ρ(ζ)

)′dζ =

∫ ξ++1

ξ+

u′(ζ)

ζ − u(ζ)dζ , but this time the

left-hand side integral diverge. It follows that ζ − u(ζ) → 0 as ξ ↓ ξ+ , whence ξ+ = ξ0 .

We see that in this case there exists a unique ξ0 such that u(ξ0) = ξ0 , and we have

ρ(ξ0) = 0 and 0 ≤ u′(ξ0) < 1 ; besides, u′ , ρ′ are different from 0 on IR \ ξ0 .

c) η− < η+ Actually, we will show that this case is impossible. Indeed, from (15) we have

ρ = ((ξ − u)ρ)′ in D′(IR) . Therefore

∫ η+

η−

ρ(ζ) dζ =[(ξ − u(ξ))ρ(ξ)

]∣∣∣η+

η−= 0 , so that

ρ|(η−,η+) ≡ 0 . Hence u′|(η−,η+) ≡ const by (16); taking into account that u(η±) = η± , we

find u′(η± ∓ 0) = 1 . Besides, 0 ≤ u′(η± ± 0) < 1 in all cases; arguing as in case a), we get

a contradiction with (17).

We conclude that ρ(·), u(·) satisfy (11),(12) and have all continuity and structure prop-

erties in Lemma 1. The monotony of ρ, u trivially implies that (13) is satisfied if and only if

self-similar ρ(·, ·), u(·, ·) satisfy (10).

IV) Conversely, (11)-(13) together with the continuity of ρ(·), u(·), u′(·) and (· − u(·))ρ′(·)imply that (ρ, u)(t, x) := (ρ, u)(x/t) is a solution of (1),(2). Indeed, (14) is straightforward.

Besides, (11),(12) yield the monotony of ρ(·), u(·) at ±∞ . Thus (10) holds also. ⋄

Using the results of Lemma 1, set ρ0 := ρ(ξ0) and k := u′(ξ0) ; define σ := ρ0 − k .

Note that σ ∈ (−1,+∞) and

ρ0 = (σ)+ = maxσ, 0, k = (σ)− = max−σ, 0. (19)

Further, split IR into the three (may be empty) intervals (−∞, ξ−) , (ξ−, ξ+) , (ξ+,∞) . Let

us inverse ρ(·) on (−∞, ξ−) and (ξ+,+∞) . We will ambiguously use the same notation

for the function ρ(·) and the independent variable ρ ∈ IR+ . The functions

ρ−1− : I(ρ0, ρ−) 7→ (−∞, ξ−), ρ−1

+ : I(ρ0, ρ+) 7→ (ξ+,+∞)

are well defined. Set

Πε±(ρ; σ) :=

∫ ρ

(σ)+

[ρ−1± (r)− u(ρ−1

± (r))]2dr − C (20)

for ρ ∈ I(ρ0, ρ±) , where C is taken from (11). Let use the simplified notation Π±(·) for

Πε±(·; σ) whenever ε, σ are fixed; besides, let ˙ denote derivation with respect to ρ . We

can rewrite (11) under the form

εu′(ξ) = p(ρ(ξ))− Π±(ρ(ξ)). (21)

Since 0 /∈ I(ρ0, ρ±) and u′(ξ) is shown to be non-zero on (−∞, ξ−)∪(ξ+,+∞) , we deduce

that Π± ∈ C2(I(ρ0, ρ±)) and that one has

ρΠ± + 2Π± =2εΠ±

f − Π±, Π± > 0 and sign(p−Π±) = sign(ρ± − ρ0) (22)

on I(ρ0, ρ±) . Further, Π± can be extended to I(ρ0, ρ±) by continuity. By (21) and

Lemma 1 Π±(·) admits finite limits at ρ0 , and we can assign

Π±(ρ0) = p(ρ0)− εk, (23)

where ρ0, k are defined by (19). Besides,

Π±(ρ±) = p(ρ±). (24)

Indeed, the right-hand side in (21) admits finite limits as ρ(ξ) → ρ± , because, in case

ρ0 < ρ± , Π±(·) are increasing and bounded from above and, in case ρ0 > ρ± , Π±(·) are

concave and bounded from below. We deduce u′(ξ) → 0 as ξ → ±∞ , since u has finite

limits at ±∞ .

In addition, in order to cover the case of ξ− = −∞ (i.e., ρ0 = ρ− ) and/or ξ+ = +∞(i.e., ρ0 = ρ+ ), we just define Π−(·) and/or Π+(·) by (24).

Finally, (20) and (12) yield u′(ξ) = ±√

Π±(ρ)ρ′(ξ)

ρfor all ξ ∈ (−∞, ξ−) and all

ξ ∈ (ξ+,+∞) , respectively. Since u(ξ−) = u(ξ+) , (13) yields that

u+ − u− =

∫ ρ+

ρ0

√Π+(r)

dr

r+

∫ ρ−

ρ0

√Π−(r)

dr

r(25)

and the integrals in the right-hand side of (25) are finite.

We have established the following result.

Proposition 1 Let (ρ, u) be a solution of (1),(2) in the sense of Definition 2. Then there

exist σ ∈ (−1,+∞) , ρ0 ∈ [0,+∞] , k ∈ [0, 1) , and functions Π± ∈ C(I(ρ0, ρ±)) ∩C2(I(ρ0, ρ±)) such that (19) and (22)-(25) hold.

The converse assertion is also true.

Proposition 2 Let σ ∈ (−1,+∞) and ρ0, k be defined by (19). Let Π± ∈ C(I(ρ0, ρ±))∩C2(I(ρ0, ρ±)) satisfy (22)-(25). Then there exists a solution (ρ, u) of (1),(2) in the sense

of Definition 2, and it is given by

ρ(t, x) = ρ(x/t) =

[Ξε

−]−1(x/t), x/t < ξ+

[Ξε+]

−1(x/t), ξ− < x/t≡

[Ξε−]

−1(x/t), x/t < ξ−

ρ0, ξ− < x/t < ξ+

[Ξε+]

−1(x/t), ξ+ < x/t,

(26)

u(t, x) = u(x/t) =

Uε− [Ξε

−]−1(x/t), x/t < ξ−

Uε−(ρ0) = Uε

+(ρ0), ξ− < x/t < ξ+

Uε+ [Ξε

+]−1(x/t), ξ+ < x/t,

(27)

where

Uε±(ρ) := u± ∓

∫ ρ±

ρ

√Π±(r)

dr

rfor ρ ∈ I(ρ0, ρ±), (28)

Ξε±(ρ) := Uε

±(ρ)±√Π±(ρ) for ρ ∈ I(ρ0, ρ±), (29)

and ξ± are defined by

ξ± := limρ∈I(ρ0,ρ±), ρ→ρ0

Ξε±(ρ) if ρ0 6= ρ±;

ξ− := −∞ and/or ξ+ := +∞ if ρ0 = ρ− and/or ρ0 = ρ+.

(30)

Proof: The cases ρ0 = ρ+ , ρ0 = ρ− are trivial; assume ρ0 6= ρ± . Let Uε±(·) , Ξε

±(·)be defined by (28),(29). Note that both integrals in (28) necessarily converge as ρ → ρ0 ,

ρ ∈ I(ρ0, ρ±) . Indeed, in case ρ0 = 0 both are positive and thus finite, by (25). In case

ρ0 > 0 , Π± are bounded as ρ → ρ0 , by monotony of ρ2Π± which is evident from (22).

Therefore Uε±(ρ0) are well defined; by (25) they coincide. Note also that

Ξε±(ρ) = ±ρΠ±(ρ) + 2Π±(ρ)

2ρ√

Π±(ρ)= ±

ε√Π±(ρ)

ρ(p(ρ)−Π±(ρ))(31)

are continuous and non-zero on I(ρ0, ρ±) , so that ξ± and [Ξε±]

−1 are well defined in the

graph sense. We see from (25),(28),(29) that

ξ+ − ξ− = limρ∈I(ρ0,ρ−), ρ→ρ0

√Π−(ρ) + lim

ρ∈I(ρ0,ρ+), ρ→ρ0

√Π+(ρ) ≥ 0. (32)

Clearly, ρ, u ∈ C(IR)∩C1(IR\ξ−, ξ+) ; besides, (26)-(29),(31) and (24) yield (13),(21)

for all ξ 6= ξ± . In fact, we could show that (24),(22) imply that√

Π±(ρ) → +∞ as

ρ → ρ± , ρ ∈ I(ρ0, ρ±) , so that Ξε± map I(ρ0, ρ±) onto (ξ+,+∞) and (−∞, ξ−) ,

respectively. Still (24) yields (21) for ξ outside the range of Ξε± even if we admit that Ξε

±

can be bounded. Futhermore, by (31),(21),(29) and (26),(27) we have for all ξ 6= ξ±

(ξ − u(ξ))ρ′(ξ) = ±ρ(p(ρ(ξ))−Π±(ρ(ξ)))

±ε = ρ(ξ)u′(ξ); (33)

whence (12) follows for ξ 6= ξ± .

Now consider the two possibilities:

a) ρ0 > 0 . Then u′(ξ± ± 0) = 0 by (21) and (24), while in case ξ− < ξ+ we have

u′(ξ± ∓ 0) = 0 by (27). Thus u ∈ C1(IR) and, by (33), (· − u(·))ρ′(·) is continuous on

IR . It now follows that (11),(12) hold everywhere; besides, (13) is obvious.

b) ρ0 = 0 . Then Π±(ρ) → 0 as ρ → ρ0 , ρ ∈ I(ρ0, ρ±) . Indeed, by (22) we have for

ρ ∈ I(ρ0, ρ±)

∫ (ρ0+ρ+)/2

ρ

(ln Π±(r)

)·dr =

∫ (ρ0+ρ+)/2

ρ

2

r

( ε

p(r)−Π±(r)− 1)dr.

By (23),(19) it follows that ε/(p(r)− Π±(r)) − 1 → 1/k − 1 > 0 as ρ → ρ0 . Therefore

the last integral diverge, hence ln Π±(ρ) → −∞ as ρ → ρ0 , ρ ∈ I(ρ0, ρ±) . We conclude

by (32) that ξ− = ξ+ ; by (21) and (23), u′(ξ± ± 0) = k . Thus again u′(·) , (· − u(·))ρ′(·)are continuous on IR , and (11)-(13) hold.

By Lemma 1, it follows that (ρ, u) is a solution of (1),(2). ⋄

2 Existence and uniqueness of viscous approximations

In Section 1, we have reduced the problem (1),(2) to finding σ ∈ (−1,+∞) , ρ0 , k and a

pair of functions Π± ∈ C(I(ρ0, ρ±))∩C2(I(ρ0, ρ±)) that satisfy (19) and (22)-(25). In this

section we prove that such σ, ρ0, k,Π±(·) do exist (and are unique), provided p(·) satisfies

(9) (cf. Proposition 3 and Lemmae 4,5,6 below). We also prove two preliminary convergence

results (cf. Lemma 3 and Proposition 4).

Start by fixing σ ∈ [−1,+∞) and b ∈ (0,+∞) . Set a := (σ)+ , k := (σ)− and

consider the problem of finding Π ∈ C(I(a, b) ∩ C2(I(a, b)) satisfying

ρΠ(ρ) + 2Π(ρ) =2εΠ(ρ)

p(ρ)−Π(ρ)with Π(ρ) > 0 and (b− a)(p(ρ)− Π(ρ)) > 0

for all ρ ∈ I(a, b),

Π(a) = p(a)− εk,

Π(b) = p(b).

(34)

Proposition 3 There exists a unique solution to the problem (34).

Let Πε(·) denote the solution of (34) corresponding to ε , ε > 0 . Recall Definition 1.

1.III.2. Existence and uniqueness of viscous approximations 73

Proposition 4 As ε ↓ 0 , Πε converge to the (-1)-hull of p(·) uniformly on I(a, b) .

First note that the equation in (34) is still in the scope of the maximum principle (cf.

Chapter 1.II). For the sake of completeness, we restate it here for the case a < b .

Lemma 2 [Maximum Principle] Let Π,Υ ∈ C[a, b]∩C2(a, b) and satisfy, for all ρ ∈ (a, b) ,

the equations Π(ρ) = G(ρ,Π(ρ), Π(ρ)) and Υ(ρ) = H(ρ,Υ(ρ), Υ(ρ)) , respectively, with

some G,H : (a, b)× IR× (0,+∞) 7→ (0,+∞] .

a) Assume that G(ρ, z, w) < H(ρ, ζ, w) for all ρ ∈ (a, b) such that Π(ρ) < Υ(ρ)

and all z, ζ, w such that z < ζ . Then Π ≥ Υ on [a, b] whenever Π(a) ≥ Υ(a) and

Π(b) ≥ Υ(b) .

b) Assume that G(ρ, z, w) ≡ H(ρ, z, w) , increases in w and strictly increases in z ; let

Π(a) = Υ(a) or Π(b) = Υ(b) . Then (Π−Υ) is monotone on [a, b] .

Proof: The proof is straightforward. ⋄

Secondly, note the following lemma, which will also be an ingredient of the convergence

proof in Section 3. Recall (5); as in Definition 1, we understand 1/a as +∞ in case a = 0 .

Lemma 3 Let [a, b] ⊂ IR+ and P εε≥0 ⊂ C[a, b] be a set of functions such that F ε =

T−1P ε are concave on (1/b, 1/a) . Assume that, for all ρ ∈ [a, b] , P ε(ρ) converge to

P 0(ρ) as ε ↓ 0 . Then the following assertions hold.

(a) This convergence is uniform on each segment [c, d] ⊂ (a, b) , and for each ε , P ε

is absolutely continuous on all segment [c, d] ⊂ (a, b) . Moreover, P ε are bounded

uniformly in ε a.e. on all segment [c, d] ⊂ [a, b) such that c > 0 .

(b) For all sequence εn ↓ 0 , P εn → P 0 a.e. on (a, b) ; the convergence take place

everywhere on (a, b) in case P εn, P 0 ∈ C1(a, b) .

(c) Let P ε be increasing and

Ξε±(ρ) := const∓

∫ b

ρ

√P ε(r)

dr

r±√P ε(ρ). (35)

Then Ξε±(·) are a.e. defined monotone functions on (a, b) , so that [Ξε

±]−1 are well

defined in the graph sense.

(d) For P ε increasing, one has

∫ b

a

√P εn(r)

dr

r→∫ b

a

√P 0(r)

dr

rfor all sequence εn ↓

0 , uniformly in ρ ∈ [c, d] for all [c, d] ⊂ [a, b] such that c > 0 .

(e) With the notation of (c), for all sequence εn ↓ 0 one has [Ξεn+ ]−1(ξ) → [Ξ0

+]−1(ξ) for

all ξ such that [Ξ0+]

−1 is continuous at the point ξ . The same holds with [Ξεn− ]−1 ,

[Ξ0−]

−1 in the place of [Ξεn+ ]−1 , [Ξ0

+]−1 .

Analogous properties hold if P ε are defined on [b, a] ⊂ IR+ \ 0 and F ε = T−1P ε are

convex on (1/a, 1/b) .

Proof: Since F ε = T−1P ε are concave, they are differentiable a.e. Moreover, the conver-

gence of P ε(ρ) to P 0(ρ) implies that F ε(1/ρ) → F ε(1/ρ) as ε ↓ 0 . Therefore F ε → F 0

uniformly on all segment [1/d, 1/c] ⊂ (1/b, 1/a) , and d/dV F εn → d/dV F 0 a.e. on

(1/b, 1/a) as ε ↓ 0 . Since P ε(ρ) = 1/ρ2 d/dV F ε(V ) whenever d/dV F ε(V ) exists, (a)

and (b) are evident.

Further, d/dV F ε ≥ 0 in case (c). Substituting v = 1/r in the integral in (35), we

obtain

Ξε±(ρ) = const±

∫ 1/ρ

1/b

(√d

dVF ε(1/ρ)−

√d

dVF ε(v)

)dv ± 1/b

√d

dVF ε(1/ρ),

which is monotone because d/dV F ε is monotone; hence (c). Besides, (d) follows from the

continuity and convergence of F ε(·) at V = 1/b . Indeed, one has for ρ > c > 0

∣∣∣∫ b

ρ

√P εn(r)

dr

r−∫ b

ρ

√P 0(r)

dr

r

∣∣∣ ≤∫ 1/ρ

1/b

∣∣∣√

d

dVF εn(v)−

√d

dVF 0(v)

∣∣∣ dv. (36)

Take δ > 0 and integrate |√d/dV F εn(V )−

√d/dV F 0(V ) | separately over (1/b, 1/b+δ)

and (1/b + δ, 1/c) . For all δ > 0 , the second integral vanishes as εn ↓ 0 , by (a) and the

Lebesgue theorem. Besides, the first one can be made as small as desired by choosing δ small

enough, because one has

∫ 1/b

1/b+δ

√d

dVF εn(v) dv ≤

∫ 1/b

1/b+δ

(1 +d

dVF εn(v)) dv = δ + (F εn(1/b+ δ)− F εn(1/b)) ≤

≤ 2δ + (F 0(1/b+ δ)− F 0(1/b))

for εn sufficiently small. Hence the left-hand side of (36) can be made as small as desired

uniformly in ρ ∈ [c, d] . Moreover, if a > 0 , we can take c = a in the reasoning above and

prove (d).

Finally, (b) and (d) imply that Ξεn± → Ξ0

± a.e. on (a, b) . It is classical in the basic

probability theory that the a.e. convergence of monotone functions (interpreted as random

variables) implies the pointwise convergence of their graph inverse functions (interpreted as

their distribution functions) at the points of continuity of the limit (e.g., cf. [Sv]). Thus (e)

follows.

The case of convex F ε is similar. ⋄

Proof of Proposition 3: The case a = b is trivial; for definiteness, assume a 0 we could prove the existence directly by penalisation of the right-hand side of (34),

applying Lemmae 2,4 and following the corresponding proof in Chapter 1.II. Instead, we will

perform the transformation T−1 : [ρ ∈ (a, b) 7→ Π(ρ)] 7→ [V ∈ (1/b, 1/a) 7→ −Π(1/V )] . It

reduces the equation in (34) to the equation

d2

dV 2Φ(V ) =

1

V

2ε d/dV Φ(V )

f(V )− Φ(V )(37)

with d/dV Φ > 0 and Φ > f on (1/b, 1/a) , where Φ = T−1Π and f = T−1p . This

equation only differs from the one that appears in Chapter 1.II by the factor 1/V in the

right-hand side. This factor is continuous and bounded on (1/b, 1/a) since a > 0 , so

that the proof in Chapter 1.II applies without any further modification. Hence there exists

a strictly increasing concave solution Φ ∈ C[1/b, 1/a] ∩ C2(1/b, 1/a) to the problem (37),

Φ(1/a) = f(1/a) , Φ(1/b) = f(1/b) . Therefore Π = TΦ ∈ C[a, b] ∩ C2(a, b) and Π

verifies (34).

For a = 0 , let first find, for all δ ∈ (0, b) a function Πδ ∈ C[0, b] ∩ C2(δ, b) such that

ρΠδ + 2Πδ =2εΠδ

p− Πδ

with Πδ > 0, p−Πδ > 0 on [δ, b);

Πδ(b) = p(b), Πδ|[0,δ] = −kε.

The proof of existence goes on as above. By Lemma 2(b), there exists a function Π on

[0, b] such that Πδ ↑ Π as δ ↓ 0 . Furthermore, applying one more time the same proof

from Chapter 1.II, this time to the functions T−1Πδ , we infer that T−1Π ∈ C2(1/d, 1/c)

and T−1Π satisfies (37) on (1/d, 1/c) for all segment [1/d, 1/c] ⊂ (1/b,+∞) . Thus

Π ∈ C2(0, b) , and the equation in (34) holds. Besides, the continuity of Π at 0 and b

follows from Lemma 2(a) by comparison with special solutions of the equation in (34), as in

Chapter 1.II. Thus Π(0) = −kε and Π(b) = p(b) by the construction of Πδ . ⋄

Proof of Proposition 4: Let us adapt the proof from Chapters 1.I,1.II. Let a 0 and construct a barrier function

Υα ∈ C2(a, b) ∩ C[a, b] such that ρΥα + 2Υα ≥ m(α) > 0 and α/2 ≤ P − Υα < α

on (a, b) . Then apply Lemma 2(a) to Πε and Υα . By Lemma 3(a) 1/ρ Πε is uniformly

bounded on all segment [c, d] ⊂ (a, b) . It follows that P ≥ Πε ≥ Υα on [a, b] for all ε

sufficiently small. ⋄

Finally, note the following property of solutions of (34).

Lemma 4 Let Π(·; σ) verify (34) with a = 0 , k = −σ ∈ [0, 1) . Then Π(ρ; σ) → 0 as

ρ → 0 , and the integral S(σ) =

∫ b

0

√Π(r; σ)

dr

ris finite. In addition, S(σ) ↑ +∞ as

σ ↓ −1 .

Proof: Clearly, there is no problem in convergence of the integral on the upper limit. Let

prove that S(σ) converge at the lower limit. Set κ = 1− (1− k)/2 ∈ [1/2, 1) . Since Π, p

are continuous on [0, b] and p(0)−Π(0; σ) = kε , there exists δ > 0 such that p−Π ≤ κε

on (0, δ) . By (34), one has on (0, δ) Π(δ; σ) ≥ Π(ρ; σ)+

∫ δ

ρ

( εκε

− 1)2rΠ(r; σ) dr .

Hence Π(ρ; σ) ≥ const ρ−2(1/κ−1) by the Gronwall inequality, so that the first two assertions

are evident.

Now consider the function Π(·;−1) . If S(−1) diverge, then limσ↓−1 S(σ) = +∞ ,

by Lemma 2(b) and the Levi theorem. Assume that S(−1) < +∞ ; we will arrive to a

contradiction. Indeed, there are two cases.

a) lim supρ↓0 Π(ρ; σ) = 0 . Proceeding as in Proposition 2, we can therefore define ξ+ :=

limρ↓0

(∫ ξ+

ρ

√Π+(r;−1)

dr

r+

√Π+(ρ;−1)

)= S(−1) . Introduce the C1 functions r, u :

ξ ∈ [ξ+,+∞) 7→ ρ(ξ), u(ξ) by formulae (26),(27) and rewrite (34) for ξ, ρ, u . This yields

ρ(ξ+) = 0 , ξ+ − u(ξ+) = 0 , u′(ξ+ + 0) = 1 and

ε(u′(ξ)− u′(ξ+ + 0)) = −∫ ξ

ξ+

(ζ − u(ζ))ρ(ζ)u′(ζ) dζ + p(ρ(ξ)).

for all ξ > ξ+ . As it is shown in part III of the proof of Lemma 1, these properties are

incompartible.

b) lim supρ↓0 Π(ρ; σ) = l2 > 0 . Take κ ∈ (0, 1) . There exists δ0 ∈ (0, b) and σ0 ∈(−1/2,−1) such that p(ρ) − Π(ρ; σ0) ≥ κε for all ρ ∈ [0, δ0] . Take δ1 ∈ (0, δ0] such

that Π(δ1;−1) ≥ l2/2 . Note that, by Lemma 2(a), Π(·; σ) ↓ Π(·;−1) as σ ↓ −1 .

By Lemma 3(b), we also have Π(δ1; σ) → Π(δ1;−1) as σ ↓ −1 . Therefore there exists

σ1 ∈ [σ0,−1) such that Π(δ1; σ1) ≥ l2/4 ; in addition, p(ρ) − Π(ρ; σ1) ≥ κε for all

ρ ∈ [0; δ1] . Now (34) yields Π(δ1; σ1) ≤ Π(ρ; σ1) +

∫ δ1

ρ

( εκε

− 1)2rΠ(r; σ1) dr for all ρ ∈

[0, δ1] . Applying the Gronwall inequality and performing easy calculations, we get S(σ1) ≥∫ δ1

0

√Π(r; σ1)

dr

r≥ l

2δ1/κ−1 δ

1−1/κ

1/κ− 1=

l

2

κ

1− κ. Letting κ ↓ 1 , we conclude that S(-

1) = limσ↓−1 S(σ) = +∞ , which contradicts our assumption. ⋄

For σ ∈ (−1,+∞) denote by Sε±(σ) the integrals

∫ ρ±

(σ)+

√Πε

±(r; σ)dr

r, where Πε

±(·; σ)denotes the unique solution of (22)-(24), according to Proposition 3. By Lemma 4, Sε

±(σ)

are finite for σ ∈ (−1, 0] ; evidently, it is also true in case σ ∈ (0,+∞) .

The next step consists in varying σ in order to satisfy the condition (25), i.e.,

u+ − u− = Sε+(σ) + Sε

−(σ),

which is now shown to be meaningful. We will prove that, for ρ± and ε fixed, (25) establishes

a bijection between σ ∈ (−1,+∞) and u+ − u− ∈ IR , provided p(·) satisfies (9). For

ρ0 = (σ)+ ∈ [0,+∞) , denote by P±(·; ρ0) the (-1)-hulls of p(·) on I(ρ0, ρ±) , respectively.

It will be convenient to extend Πε±(·; σ) , P±(·; ρ0) to continuous functions on IR+ , by

setting them constant on each component of IR+ \ I(ρ0, ρ±) . As in Chapter 1.II, we have

the following two results.

Lemma 5 With the notation above and σ ∈ (−1,+∞), ρ0 = (σ)+ , the following holds.

(a) For all ρ ∈ IR+ and ε > 0 , σ 7→ Πε±(ρ; σ) do not decrease; nor do ρ0 7→ P±(ρ; ρ0) .

(b) For all ρ ∈ IR+ and ε > 0 , σ 7→ sign(ρ±−ρ0)Πε±(ρ; σ) do not increase; nor do

σ 7→ sign(ρ±−ρ0)P±(ρ; ρ0) .

(c) For all ε > 0 the maps σ 7→ Πε±(·; σ) are continuous for the L∞(IR+) topology; so

do σ 7→ P±(·; ρ0) .

(d) For all ε > 0 , the functions σ 7→ Sε±(σ) are continuous and strictly decreasing; so do

the functions ρ0 7→∫ ρ±

ρ0

√P±(r; ρ0)

dr

r.

Proof: The properties (a)-(c) for ε > 0 follow from Lemma 2. Hence (a)-(c) for P±

follow by Proposition 4 and Lemma 3(b). Now (b) and the Levi theorem yield (d). ⋄

Lemma 6 Let ρ±, u± be fixed.

(a) Assume (9)(a) holds. Then for all ε > 0 one has Sε±(σ) → −∞ as σ → +∞ .

(b) Assume (9)(b) holds. Then for all L > 0 there exists ε0(L) > 0 such that for all

0 < ε < ε0(L) one has limσ→+∞Sε±(σ) < −L .

(c) In both cases, one has

∫ ρ±

ρ0

√P±(r; ρ0)

dr

r→ −∞ as ρ0 → +∞ .

Proof: The proof is much the same as the one of Proposition 4 in Chapter 1.II.

(a) Assume, for instance, that S+(σ) is bounded from below by some constant −M ∈IR− . Set V0 := 1/(σ)+ , V+ := 1/ρ+ > V0 , and perform the transformation T−1p = f ,

T−1Πε+(·; σ) = Φ(·;V0) . We have Φ(·;V0) which is convex on [V0, V+] , Φ(V0;V0) =

f(V0) , Φ(V+;V0) = f(V+) , and Φ(·;V0) satisfies the equation (37) on (V0, V+) . Note that

f(V0) → −∞ as V0 → −∞ , by (9)(a). Therefore the convexity of Φ(·;V0) implies that

Φ(V+/2;V0) → −∞ as V0 → 0 ; on the other hand, it yields (V+/2)√d/dV Φ(V+/2;V0) ≤

−Sε+(σ) ≤ M independently of V0 ∈ (0, V+/2) . Hence we have a family of functions

Φ(·;V0)V0∈(0,V+/2) such that Φ(V+/2;V0) ↓ −∞ as V0 ↓ 0 , Φ(V+/2;V0) ≤ (2M/V+)2

uniformly in V0 , Φ(V+;V0) = f(V+) , and Φ(·;V0) satisfy (37) on (V+/2, V+) . Let us

compare Φ(·;V0) to the solution Ψ of the Cauchy problem Ψ(V+/2) = Φ(V+/2;V0) ,

Ψ(V+/2) = (2M/V+)2 + 1 for (37). By the maximum principle Lemma 2(b), which is valid

for the equation (37), we arrive to the conclusion that Φ(V+;V0) < Ψ(V+) , while this last

value can be made well-defined and strictly less than f(V+) for V0 small enough. This

contradiction proves (a).

For the proof of (c) in this case it is sufficient to pass to F±(·;V0) = T−1P±(·; ρ0) and

note that

∫ ρ±

ρ0

√P±(r; ρ0)

dr

r= −

∫ V±

V0

√d/dV F±(v;V0) dv . The last integrals diverge as

V0 ↓ 0 ; it is easy to see because F±(·;V0) are convex, F±(V0;V0) = f(V0) → −∞ and√d/dV F±(V ;V0) ≥ min1, d/dV F±(V ;V0) .

(b) First assume that f = T−1p has a finite number of points of inflexion. By Definition 1,

the function F+(·;V0) = T−1P+(·; ρ0) is the convex hull of f(·) on [V0, V+] = [1/ρ0, 1/ρ+] .

Its graph consists of a finite number of fragments of the graph of f(·) and of chords of this

graph. It follows easily by the Cauchy-Schwarz inequality that

∫ ρ+

ρ0

√P+(r; ρ0)

dr

r= −

∫ V+

V0

√d/dV F+(v;V0) dv ≤

≤ −∫ V+

V0

√d/dV f(v) dv =

∫ ρ+

ρ0

√p(r)

dρ

ρ→ −∞

as ρ0 → +∞ , by (9)(b). It follows by a density argument that

∫ ρ+

ρ0

√P+(r; ρ0)

dr

r→ −∞

as ρ0 → +∞ for a general flux function p(·) , which ends the proof of (c). Besides, the

claim of (b) follows now by Proposition 4 and Lemma 3(d). ⋄

Combining Propositions 1,2 and 3 and Lemmae 4,5,6, one easily deduces the main result

of this section.

Theorem 1 Let p(·) be continuous and strictly increasing on IR+ .

(a) Assume (9)(a) holds. Then for all ρ± > 0 , u± ∈ IR , ε > 0 there exists a unique

solution to the problem (1),(2) in the sense of Definition 2.

(b) Assume (9)(b) holds. Then for all ρ± > 0 , u± ∈ IR there exists ε0 = ε0(ρ±, u+−u−)such that for all 0 < ε < ε0 there exists a unique solution to the problem (1),(2) in

the sense of Definition 2.

Remark 2 At the present stage, it is not clear to the author whether the condition (9)(b)

suffices to have the existence of solutions to (1),(2) for all ε > 0 . On the other hand, it is

easy to see that if p(·) is locally absolutely continuous but fails to satisfy (9)(b), there is no

existence of a bounded self-similar solution to (1),(2) for arbitrary Riemann data. The exact

bound on u+ − u− in order to have the existence for ε in some non-empty neigbourhood of

0 is

−(u+ − u−) < limρ0→+∞

∫ ρ0

ρ+

√P+(r; ρ0)

dr

r+ lim

ρ0→+∞

∫ ρ0

ρ−

√P−(r; ρ0)

dr

r, (38)

where P±(·; ρ0) are the (-1)-hulls of p(·) on [ρ+, ρ0] and [ρ−, ρ0] , respectively. This

assertion follows from the extremal property of the upper (-1)-hull P (·) on [a, b] ⊂ (0,+∞) ,

which in fact maximizes the integral

∫ b

a

√Q(r)

dr

rwith respect to all Q(·) that satisfy the

assumptions (7)(i),(ii) in Definition 1.

1.III.3. Admissible solution of the problem (8),(2) 79

3 The admissible solution of the problem (8),(2)

In this section we study the global (with respect to data ρ±, u± ) solvability for the system

(8),(2) subject to the wave fan admissibility criterion (cf. [D74, D89, Tz96]).

Definition 3 Let (ρ, u) be a pair of functions such that ρ ∈ L∞(IR+ × IR; IR+) and

u ∈ L∞(ρ > 0; IR) , where ρ > 0 := (t, x) ∈ IR+ × IR | ρ(t, x) > 0 , ρ being an a.e.

defined representative of ρ . Then (ρ, u) is a wave-fan admissible solution of the problem

(8),(2), if

(i) the equations ρt + qx = 0 , qt + (e + p(ρ))x = 0 are fulfilled in D′(IR+ × IR) , where

q = ρu, e = ρu2 on the set ρ > 0 and q = 0, e = 0 on its complementary;

(ii) one has ess limt↓0

(‖ρ(t, ·)− ρ(0, ·)‖L1(−R,R) + ‖u(t, ·)− u(0, ·)‖L1(−R,R)

)= 0

for all R > 0 , where ρ(0, ·), u(0, ·) are given by (2);

(iii) in addition, there exists a sequence εnn∈IN ⊂ (0,+∞) , εn → 0 as n → ∞ , such

that (ρεn , uεn) tends to (ρ, u) as n → ∞ , where (ρεn , uεn) is a solution of (1),(2)

in the sense of Definition 2; more exactly, this means that ρεn → ρ , ρεnuεn → q and

ρεn(uεn)2 → e a.e. on IR+ × IR .

Note that, according to (i) of Definition 3, u remains undefined within the vaccuum state.

While (i),(ii) define a weak solution, (iii) presents an additional selection criterion which is

stronger than (i) unless p(·) degenerates to p(ρ) = const/ρ on some interval of IR+

depending on ρ±, u± . If such a degeneration does not occur, an infinity of weak solutions

that are not admissible can be observed (cf. [KrSt93]).

Due to (iii), a wave-fan admissible solution is actually self-similar. According to formulae

(40)-(45) and Theorem 2 below, it is unique and has the usual wave-fan structure. The

solution contains at most three “main” constant states: (ρ−, u−) at −∞ ; the vaccuum

state ρ ≡ 0 or a constant state in a neighbourhood of the unique point ξ0 such that there

exists limξ→ξ0 u(ξ) = ξ0 (in case there is no vaccuum state); and (ρ+, u+) at +∞ . These

states are separated by two wave fans of the first and the second family, respectively. Each

wave fan is a sequence of shocks, rarefactions, contact discontinuities and (in case p(·) is

not smooth) “rarefaction type” constant states. There is no vaccuum state in the solution

unless the intermediate “main” state is the one. In this case, u(ξ) − ξ → 0 as ξ enters

the vaccuum state from any side (cf. (41),(43) and (48) below). The necessary and sufficient

condition for vaccuum state to appear is that

u+ − u− ≥∫ ρ+

0

√P+(r; 0)

dr

r+

∫ ρ−

0

√P−(r; 0)

dr

r, (39)

where P±(·; 0) are the (-1)-hulls defined below. All these properties can be observed from

the following formulae for the wave-fan admissible solution, suggested by Propositions 2 and

4:

ρ(t, x) = ρ(x/t) =

[Ξ−]

−1(x/t), x/t < ξ+

[Ξ+]−1(x/t), ξ− < x/t

≡

[Ξ−]−1(x/t), x/t < ξ−

ρ0, ξ− < x/t < ξ+

[Ξ+]−1(x/t), ξ+ < x/t,

(40)

u(t, x) = u(x/t) =

U− [Ξ−]−1(x/t), x/t < ξ−

U−(ρ0) = U+(ρ0), ξ− < x/t < ξ+

(in case ρ0 > 0)

U+ [Ξ+]−1(x/t), ξ+ < x/t,

(41)

where

U±(ρ) := u± ∓∫ ρ±

ρ

√P±(r; ρ0)

dr

rfor ρ ∈ I(ρ0, ρ±), (42)

Ξ±(ρ) := U±(ρ)±√P±(ρ; ρ0) for ρ ∈ I(ρ0, ρ±), (43)

ξ± are defined by

ξ± := limρ∈I(ρ0,ρ±), ρ→ρ0

Ξ±(ρ) if ρ0 6= ρ±;

ξ− := −∞ and/or ξ+ := +∞ if ρ0 = ρ− and/or ρ0 = ρ+,

(44)

ρ0 = 0 in case (39) holds, and ρ0 ∈ (0,+∞) is the unique value that satisfies the relation

u+ − u− =

∫ ρ+

ρ0

√P+(r; ρ0)

dr

r+

∫ ρ−

ρ0

√P−(r; ρ0)

dr

r(45)

in case (39) fails; finally, P±(·; ρ0) are the (-1)-hulls of the graph of p(·) on I(ρ0, ρ±) ,

respectively, as defined in Definition 1.

Let state the main result of this chapter.

Theorem 2 Assume that continuous strictly increasing function p(·) satisfies (9). Let ρ± >

0 , u± ∈ IR . Then the solution (ρε, uε) to the problem (8),(2) (which exists and is unique, at

least for ε sufficiently small) tends as ε ↓ 0 , in the sense of Definition 3(iii), to (ρ, u) given

by the formulae (40)-(45). The pair (ρ, u) is the unique solution of the Riemann problem

(8),(2) in the sense of Definition 3.

Proof: According to Theorem 1, there exists a unique solution to (1),(2), at least for

ε sufficiently small; let denote this solution by (ρε, uε) . By Propositions 1 and 2 (ρε, uε)

corresponds to some σε ∈ (−1,+∞) and Πε±(·; σε) such that (22)-(25) and (26)-(30)

hold. Consider the set (σε)+ ⊂ IR+ ; it has an accumulation point ρ0 in IR+ . Choose

1.III.3. Admissible solution of the problem (8),(2) 81

a sequence εn ↓ 0 such that σεn → σ0 as n → ∞ ; let omit the subscript n , because

we will prove later that σ0 does not depend on any particular sequence of ε ↓ 0 . Consider

separately the three possibilities: ρ0 ∈ (0,+∞) , ρ0 = +∞ and ρ0 = 0 .

a) ρ0 ∈ (0,+∞) . As in Chapter 1.II, from Proposition 4 and Lemma 5(a),(c) it follows

that

Πε±(·; σε) → P±(·; ρ0) in L∞(IR+) as ε → 0, (46)

where Πε±, P± are extended to IR+ as in Lemma 5. Similarly, extend Uε

± , Uε in (28)

and (42) by constants to continuous functions on IR+ . By Lemma 3(d), Uε±(·) converge to

U±(·) uniformly on I(ρ0, ρ±) , respectively. Since (25) and (45) write also as Uε−((σ

ε)+) =

Uε+((σ

ε)+) and U−(ρ0) = U+(ρ0) , one deduces (45). It follows by Lemma 5 that ρ0 is

uniquely determined by p(·) , ρ± and u± in case a).

Further, by Lemma 3(c) and (43),(44) one has Ξ±(ρ1) ≤ ξ− ≤ U−(ρ0) = U+(ρ0) ≤ξ+ ≤ Ξ+(ρ2) for all ρ1 ∈ I(ρ0, ρ−) , ρ2 ∈ I(ρ0, ρ+) . Considering separately the cases

ρ0 > ρ+, ρ0 = ρ+ , ρ0 < ρ+ , one deduces by Lemma 3(e) from (46),(26) and (40) that

ρε(·) → ρ(·) a.e. on (ξ−,+∞) . Similarly, one gets ρε(·) → ρ(·) a.e. on (−∞, ξ+) , so

that the convergence actually takes place for a.a. ξ ∈ IR . Consequently, uε(·) → uε(·) a.e.

on IR .

b) ρ0 = +∞ . Actually, this case is impossible. Indeed, for all L > 0 we have

u+ − u− =

∫ ρ+

σε

√Πε

+(r; σε)dr

r+

∫ ρ−

σε

√Πε

−(r; σε)dr

r≥

≥∫ ρ+

L

√Πε

+(r;L)dr

r+

∫ ρ−

L

√Πε

−(r;L)dr

r

for ε small enough, by Lemma 5(d). Passing to the limit as ε→ 0 , we obtain

u+ − u− ≥∫ ρ+

L

√P+(r;L)

dr

r+

∫ ρ−

L

√P−(r;L)

dr

r

by Proposition 4 and Lemma 3(d). By Lemma 6(c), this last quantity tends to −∞ as

L→ +∞ , provided (9) holds. Thus u+ − u− = −∞ , which is contradictory.

c) ρ0 = 0 . As in case a), one has (46). Therefore Πε±(·; σε) → P±(·; 0) a.e. on (0, ρ±) ,

respectively, by Lemma 3(b). It follows by the Fatou Lemma that

u+ − u− = lim infε→0

(∫ ρ+

0

√Πε

+(r; σε)dr

r+

∫ ρ−

0

√Πε

−(r; σε)dr

r

)≥

≥∫ ρ+

0

√Πε

+(r; 0)dr

r+

∫ ρ−

0

√Πε

−(r; 0)dr

r.

(47)

Note that, since c) is excluded and since we have seen that in case a) this last quantity is

necessarily greater than u+ − u− , the accumulation point ρ0 is always unique and finite.

Let prove that ρε → ρ , ρεuε → ρu and ρε(uε)2 → ρu2 on the set ξ | ρ(ξ) > 0 , and

that ρε → 0 , ρεuε → 0 and ρε(uε)2 → 0 on its complementary.

First assume (47) holds with equality. Then we still have U−(0) = U+(0) . Moreover,

P (ρ; 0) → 0 as ρ→ 0 (48)

whenever vaccuum appears. Indeed, one can show as in the proof of Lemma 3(c) that

Ξ±(ρ) = u± ∓∫ ρ±

ρ

√P±(r; 0) ±

√P±(ρ; 0)

dr

rare monotone on (0, ρ±) ; on the other

hand,

∫ ρ±

ρ

√P±(r; 0)

dr

rare also monotone and converge as ρ ↓ 0 . Therefore there exist

limits as ρ ↓ 0 of

√P±(ρ; 0) , which are necessarily zero. Thus Ξ−(ρ1) ≤ ξ− = U−(0) =

U+(0) = ξ+ ≤ Ξ+(ρ2) for all ρ1 ∈ [0, ρ−] , ρ2 ∈ [0, ρ+] ; moreover, ρ(ξ) > 0 for all

ξ 6= ξ− = ξ+ . The convergence of ρε, uε to ρ, u , respectively, a.e. on IR follows as in case

a).

Secondly, assume that the inequality in (47) is strict. By (48) we have Ξ−(ρ1) ≤ ξ− =

U−(0) < U+(0) = ξ+ ≤ Ξ+(ρ2) , with ρ1 , ρ2 as above. One more time, we deduce

the convergence of ρε, uε to ρ, u , respectively, a.e. on (−∞, ξ−) ∪ (ξ+,+∞) . Note

that Ξ±(ρ) are strictly monotone at ρ = 0 , since P±(·; 0) are strictly increasing and

P±(+0; 0) = 0 . It follows that ρ(ξ) → 0 as ξ ↑ ξ− or ξ ↓ ξ+ . For ε sufficiently small, ρε

has no points of maximum on IR , so that for all δ > 0 we infer maxξ∈[ξ−−δ,ξ++δ] ρε(ξ) =

maxρε(ξ− − δ), ρε(ξ+ + δ) . By the last observation, it follows that ρε → 0 uniformly

on [ξ−, ξ+] . Besides, uε is nondecreasing on IR for ε sufficiently small, hence uniformly

bounded, so that ρεuε → 0 and ρε(uε)2 → 0 on [ξ−, ξ+] .

Thus we have shown that (ρ, u)(·, ·) satisfies Definition 3(i),(iii). Besides, (ρ, u)(±∞) =

(ρ±, u±) by (40)-(42). Since, in addition, ρ(·) , u(·) are both monotone at ±∞ , Defini-

tion 3(ii) holds as well. ⋄

Remark 3 In the equations of isentropic gas dynamics u represents the velocity in gas and

has no physical meaning inside the vaccuum state, while the specific impulse q and specific

energy e are, naturally, both zero within vaccuum. Nevertheless, one could ask for a limit

of uε even inside the vaccuum state. In case p ∈ W 1,1(0, R) for some R > 0 and

limρ↓0 p(ρ) = 0 , we are able to prove that the unique limit of uε inside the vaccuum state

is the identity function u(ξ) = ξ . This is due to the formula (43) and the uniform in ε

convergence of√

Πε±(ρ; 0) to 0 as ρ ↓ 0 . In turn, this last property results from (48),

Proposition 4, Lemma 3(b) and the following kind of maximum principle:

maxρ∈[0,δ]

Πε±(ρ; 0) = maxΠε

±(δ; 0), supρ∈(0,δ)

p(ρ) for δ small enough,

which can be easily deduced from the equation (22). In general, we ignore whether the

convergence always take place to the same limit u(ξ) ≡ ξ within the vaccuum state; but we

still observe that u(ξ) coincides with ξ on its boundary.

CHAPTER 1.IV

L1 -Theory of Scalar Conservation Law

with Continuous Flux Function†

Introduction

We consider the Cauchy problem

∂u

∂t+ divx φ(u) = g on Q = (t, x); t ∈ (0, T ), x ∈ IRN

u(0, ·) = f on IRN(CP )

where φ : IR 7→ IRN is only assumed to be continuous and (f, g) satisfy

f = f0 + c with c ∈ IR, f0 ∈ L1(IRN) ∩ L∞(IRN),

g ∈ L1(Q), g(t, ·) ∈ L∞(IRN) for a.a. t ∈ (0, T ) and

∫ T

0

‖g(t, ·)‖∞dt <∞.(1)

A solution of (CP) will be understood in the sense of the generalized entropy solution

(g.e.s.) as introduced by S.N.Kruzhkov (cf. [K69a, K69b, K70a]). In the case of a locally

Lipschitz continuous flux function φ , there exists a unique bounded g.e.s.; this is actually

true for any (f, g) satisfying

f ∈ L∞(IRN), g ∈ L1loc(Q),

g(t, ·) ∈ L∞(IRN) for a.a. t ∈ (0, T ) and

∫ T

0

‖g(t, ·)‖∞dt <∞.(2)

For the general continuous flux function φ the situation is more delicate. Let consider the

particular case N = 2 , φ(u) =(

|u|α−1uα

, |u|β−1uβ

). It has been shown in [KP90] that if

α 6= β , α + β < 1 , then for some f ∈ L∞(IR2) the problem (CP), with g = 0 , has

a one-parameter family of different bounded g.e.s.. On the other hand, it has been shown

†This chapter will be published in [BABK].

84 Scalar Conservation Law with Continuous Flux Function

in [BK96] that if (f, g) satisfy (1), then for any α, β > 0 there exists a unique bounded

g.e.s. of (CP). In this chapter we shall improve this last result, showing (cf. Theorem 3) that

(CP) has a unique bounded g.e.s. for any (f, g) satisfying (1) according to whether the flux

function φ satisfies

There exist orthonormal vectors ξ1, . . . , ξN−1 such that

r ∈ IR 7→ ξi · φ(r) ∈ IR is nondecreasing , i = 1, . . . , N − 1.(3)

Actually, while we shall prove another uniqueness result (cf. Theorem 4), we still do not

know whether there is or is not uniqueness of bounded g.e.s. for (f, g) satisfying (1) with

any continuous flux function φ ; but we shall prove that there always exist a maximum and

a minimum bounded g.e.s. of (CP). More precisely, for any continuous flux function φ and

(f, g) satisfyingf = f0 + c with c ∈ IR, f0 ∈ L∞

0 (IRN) =h ∈ L∞(IRN); λN|h| > δ <∞† ∀δ > 0

g ∈ L1loc(Q), g(t, ·) ∈ L∞

0 (IRN) for a.a. t ∈ (0, T ) and

∫ T

0

‖g(t, ·)‖∞dt <∞,(4)

for all c ∈ IR there exist a maximum and minimum bounded g.e.s. of (CP), which coincide

except for a countable set of values of c depending on φ , f0 and g (cf. Theorem 1 and

Proposition 1).

As pointed out in [C72] and [B72], solutions of (CP) for (f, g) satisfying (1) can be

constructed through the nonlinear semigroup theory from the solutions of the equation

u+ divx φ(u) = f on IRN . (E)

As was done in [BK96], we shall derive for the equation (E) the same properties as for the

Cauchy problem (CP); actually we shall prove (cf. Corollary 1), for φ and c given, that

there is uniqueness of a bounded g.e.s. of (CP) for all (f, g) satisfying (1) if and only if there

is uniqueness of a bounded g.e.s. of (E) for all f = f0 + c with f0 ∈ L1(IRN) ∩ L∞(IRN) .

1 Existence of maximum and minimum generalized

entropy solutions

Throughout this chapter φ : IR 7→ IRN is a continuous function and we consider the Cauchy

problem (CP) as well as the equation (E). Recall the following definition:

Definition 1 Let f ∈ L1loc(IR

N) . A sub-g.e.s. (generalized entropy subsolution) (respec-

tively, super-g.e.s.) of (E) is a function u ∈ L∞loc(IR

N) satisfying

α · (u− k) + divx

α · (φ(u)− φ(k))

≤ α · (f − k) in D′(IRN) for any k ∈ IR,

† λN denote the N-dimensional Lebesgue measure; |h| > δ stands for x ∈ IRN ; |h(x)| > δ and

so on.

1.IV.1. Existence of maximum and minimum g.e.s. 85

where α = sign+(u − k) ‡ (respectively, sign−(u − k) ). A function u is a generalized

entropy solution (g.e.s.) of (E) if it is both sub- and super-g.e.s.

Definition 2 Let f ∈ L1loc(IR

N) and g ∈ L1loc(Q) . A sub-g.e.s. (respectively, super-g.e.s.)

of (CP) is a function u ∈ L∞loc(Q) satisfying

∂

∂t

α · (u− k)

+ divx

α · (φ(u)− φ(k))

≤ α · g in D′(Q) for any k ∈ IR,

where α = sign+(u−k) (respectively, sign−(u−k) ), and (u(t, ·)−f)+ → 0 (respectively,

(u(t, ·)− f)− → 0 ) in L1loc(IR

N ) as t → 0 essentially. A function u is a g.e.s. of (CP) if

it is both sub- and super-g.e.s.

The main result is the following theorem.

Theorem 1 Let (f, g) satisfy (4). Then there exist a maximum and a minimum bounded

g.e.s. of (E) and of (CP).

More precisely, considering the equation (E) and f = f0+c with c ∈ IR , f0 ∈ L∞0 (IRN) ,

we shall prove that there exists a (unique) g.e.s. u ∈ L∞(IRN) such that u ≥ u a.e. on

IRN for any sub-g.e.s. u ∈ L∞(IRN ) of (E).

This g.e.s. u will be obtained as the a.e. pointwise limit of a nonincreasing sequence

un , where un is any bounded g.e.s. of (E) corresponding to f = f0+ cn with a sequence

cn in IR decreasing to c .

The same corresponding results are valid for (CP) and minimum solutions.

The main new ingredient in the proof of Theorem 1 is the following lemma.

Lemma 1 a) Let u and u be bounded sub- and super-g.e.s., respectively, of (E), corre-

sponding to f and f ∈ L1loc(IR

N) , respectively. Assume that

λNx ∈ IRN ; u(x) > u(x)

<∞, (5)

then∫(u− u)+ +

∫

u>u

(f − f)− ≤∫

u≥u

(f − f)+, (6)

and in particular, if f ≤ f a.e. on u ≥ u , then u ≤ u a.e. on IRN .

b) Let u and u be bounded sub- and super-g.e.s., respectively, of (CP), corresponding

to (f, g) and (f , g) ∈ L1loc(IR

N)× L1loc(Q) , respectively. Assume that

λN+1

(t, x) ∈ Q; u(t, x) > u(t, x)

<∞, (7)

‡We use the notation sign+ for the Heaviside function, i.e. the characteristic function of (0,+∞) , and

sign−(r) = −sign+(−r) .

then for a.a. t ∈ (0, T )∫

(u(t, ·)− u(t, ·))+ +

∫ t

0

∫

u>u

(g − g)− ≤∫(f − f)+ +

∫ t

0

∫

u≥u

(g − g)+, (8)

and in particular, if f ≤ f a.e. on IRN and g ≤ g a.e. on u ≥ u , then u ≤ u a.e. on

Q .

Proof: Applying Lemma 3.1a) in [BK96] (cf. also [K70a, B72, C72, Ba88]), we have in the

case a)∫

(u− u)+ζ +

∫

u>u

(f − f)−ζ ≤∫

u≥u

(f − f)+ζ +

∫|φ(u)− φ(u)|χu>u |Dζ |

for all ζ ≥ 0 , ζ ∈ D(IRN) . By the assumption (5), |φ(u) − φ(u)|χu>u ∈ L1(IRN) , so

that we may let ζ tend to 1 to obtain (6) at the limit and prove a). The proof of b) is

identical using Lemma 3.1b) in [BK96] and (7). ⋄We also need the following general existence result, partially contained in [B72] and [KH74],

for which we give a complete proof in the Appendix.

Lemma 2 Let f ∈ L∞(IRN) (resp., and g ∈ L1loc(Q) satisfying g(t, ·) ∈ L∞(IRN ) for a.a.

t ∈ (0, T ) and

∫ T

0

‖g(t, ·)‖∞dt < ∞ ). Then there exists a bounded g.e.s. of (E) (resp.

(CP)).

Proof of Theorem 1: Let cn be a sequence in IR decreasing to c and, for n ∈ IN ,

un be a bounded g.e.s of (E) corresponding to fn = f0+ cn . Such a g.e.s. exists by Lemma

2.

Fix n > m . Set h = cn+cm2

and take 0 < δ < cm−cn2

; δ = cm−cn2

−α for some α > 0 .

Using Theorem 2.2a) in [BK96], we have∫(un−um+2δ)+ ≤

∫(un−h+δ)+ +

∫(h+δ−um)+ =

∫ ((un−cn)−α

)++

+

∫ ((um−cm)+α

)−≤∫(f0−α)+ +

∫(f0+α)

− =

∫(|f0| − α)+ <∞;

it follows that |un > um| <∞ and thus we deduce from Lemma 1a) that un ≤ um .

Define u = limn→∞

un ; this is, clearly, a bounded g.e.s. of (E) corresponding to f = f0+c .

Let now u be a bounded sub-g.e.s. of (E); with the same argument as above, u ≤ un a.e.

for all n and thus u ≤ u a.e. In other words, u is the maximum bounded g.e.s. of (E).

The proof of existence of the maximum bounded g.e.s. of (CP) is similar using Lemma

1b); considering a bounded g.e.s. un of (CP) corresponding to (fn, g) , we only need to

show that

supt∈[0,T ]

[∫ ((un(t)−cn)−α

)++

∫ ((um(t)−cm)+α

)−]<∞. (9)

1.IV.2. L1 semigroup approach 87

To prove (9), for M > 0 set κM (t) = inf

κ ;

∫(|g(t)| − κ)+ ≤M

. We have

κM(t) ≤ ‖g(t)‖∞ and thus κM ∈ L1(0, T ) ; on the other hand, for a.a. t ∈ (0, T ) ,

since g(t) ∈ L∞0 (IRN) , κM(t) decrease to 0 as M increase to ∞ . Thus there exists

M > 0 such that

∫ T

0

κM(t)dt ≤ α/2 . Fix M such that this is satisfied, and set kM(t) =

α/2 +

∫ t

0

κM(s)ds for t ∈ [0, T ] . Applying Theorem 2.2b) from [BK96], we get for all

t ∈ [0, T ]∫ (

(un(t)−cn)−α)+

+

∫ ((um(t)−cm)+α

)−≤∫ (

(un(t)−cn)−kM(t))+

+

+

∫ ((um(t)−cm)+kM(t)

)−≤∫ (

|f0|−α/2)+

+

∫ t

0

∫ (|g(s)|−κM(s)

)+ds ≤

≤∫ (

|f0|−α/2)+

+MT.

The proof of existence of the minimal bounded g.e.s. for (E) and (CP) is identical. ⋄We actually do not know whether there is in general uniqueness of a bounded g.e.s. of (E)

or (CP). However, we can prove the following result.

Proposition 1 Let f0 ∈ L∞0 (IRN) (resp., and g ∈ L1

loc(Q) , g(t, ·) ∈ L∞0 (IRN ) for a.a.

t ∈ (0, T ) and

∫ T

0

‖g(t, ·)‖∞dt < ∞ ). Then there exists an at most countable set N in

IR such that for all c ∈ IR \ N the equation (E) (resp., the problem (CP)) with f = f0 + c

has a unique bounded g.e.s..

Proof: For c ∈ IR , denote by u(c) (resp., u(c) ) the maximum (resp. minimum) bounded

g.e.s. By the proof above, we know that c 7→ u(c) and c 7→ u(c) are nondecreasing from

IR into L∞ continuous from the right and the left, respectively, for the L1loc topology in

L∞ ; moreover, for c1 < c2 , u(c1) ≤ u(c1) ≤ u(c2) . Thus it follows that u(c) = u(c) a.e.

for any c except an at most countable set in IR . ⋄

2 The L1 semigroup approach

In this section, using the nonlinear semigroup theory in L1 , we make the relation between

the equation (E) and the problem (CP) under the assumption (1) on the data (f, g) clearer.

For simplicity we shall assume c = 0 .

For λ > 0 and f ∈ L1(IRN) ∩ L∞(IRN) , the equation

u+ divx λφ(u) = f in IRN (10)

has a maximum bounded g.e.s. that we shall denote by J+λ f ; by Corollary 2.1 in [BK96]

J+λ f ∈ L1(IRN ) . In other words, J+

λ maps L1(IRN ) ∩ L∞(IRN) into itself. Let us start

with the following results.

Proposition 2 With the notation above, the following properties hold:

1. for any λ > 0 , J+λ is a T-contraction for the L1 -norm, i.e.

∫(J+

λ f − J+λ f)

+ ≤∫

(f − f)+ ∀f, f ∈ L1(IRN) ∩ L∞(IRN );

2. J+λ λ>0 is a resolvent family, i.e.

J+λ f = J+

µ

(µ

λf +

λ− µ

λJ+λ f

)∀λ, µ > 0, f ∈ L1(IRN) ∩ L∞(IRN);

3. the range R(J+λ ) , independent of λ by 2) , is dense in L1(IRN) .

Proof: For Part 1), let f, f ∈ L1(IRN ) ∩ L∞(IRN) and, for δ > 0 , denote by uδ, uδ

bounded g.e.s. of (E) corresponding to f + δ, f + δ respectively. As in the proof of Theorem

1, we have λNuδ > u2δ <∞ and then, by Lemma 1,

∫(uδ − u2δ)+ ≤

∫(f − f − δ)+ ≤

∫(f − f)+ . At the limit as δ → 0 , uδ → J+

λ f and u2δ → J+λ f a.e., so that we get

Part 1) by the Fatou Lemma.

For Part 2), let f ∈ L1(IRN) ∩ L∞(IRN) and assume first that λ > µ > 0 . Set

u = J+λ f ; it is a bounded g.e.s. of

v + divx µφ(v) =µ

λf +

λ− µ

λu

and so u ≤ v = J+µ

(µλf + λ−µ

λu). Then µ

λf + λ−µ

λu ≤ µ

λf + λ−µ

λv and v is a bounded

sub-g.e.s. of u + divx λφ(u) = f . We deduce v ≤ u and thus v = u . To complete the

proof of Part 2), we apply the abstract Lemma 3 below.

For the proof of Part 3), let f ∈ L1(IRN) ∩ L∞(IRN) and, for λ > 0 , set uλ = J+λ f .

We have uλ ∈ R(J+λ ) and this set is clearly, by Part 2) , independent of λ . Since ‖uλ‖∞ ≤

‖f‖∞ (see Corollary 2.1 in [BK96]), it follows immediately that uλ → f in D′(IRN) as

λ→ 0 ; indeed, being a g.e.s., uλ is also a solution of (10) in the sense of distributions. Now

using translation invariance and Part 1),

∫|uλ(x+h)−uλ(x)|dx ≤

∫|f(x+h)− f(x)|dx ,

so that the set uλλ>0 is relatively compact in L1loc(IR

N) and uλ → f in L1loc(IR

N) . At

last ‖uλ‖1 ≤ ‖f‖1 (cf. Corollary 2.1 in [BK96]) so that uλ → f in L1(IRN) ; indeed, for

any compact set K in IRN we have

lim supλ→0

‖uλ − f‖1 ≤ lim supλ→0

(∫

K

|uλ − f |+∫

|uλ| −∫

K

|uλ|+∫

IRN \K

|f |)

≤

≤∫

|f | −∫

K

|f |+∫

IRN\K

|f | = 2

∫

IRN\K

|f |,

which can be made as small as we want. ⋄

1.IV.2. L1 semigroup approach 89

Lemma 3 Let X0 be a linear subspace of a Banach space X and Jλλ>0 be a family of

non-expansive mappings from X0 into X0 . If the resolvent identity Jλ = Jµ(µλI + λ−µ

λJλ)

holds for all 0 < µ < λ , then it still holds for any λ, µ > 0 .

Proof: Following [BCP], Exercise E8.2, denote by Aλ the multivalued operator from X0

into itself defined by

v ∈ Aλu ⇔ u, v ∈ X0, u = Jλ(u+ λv);

the graph of this operator is(Jλf,

f−Jλfλ

); f ∈ X0

and one has (I + λAλ)

−1 = Jλ .

For given λ, µ > 0 , the equality Jλ = Jµ(µλI + λ−µ

λJλ)

is equivalent to the inclusion

Jλ ⊂ Jµ(µλI + λ−µ

λJλ)

since these two maps are everywhere defined in the linear space X0 ;

so it is also equivalent to the inclusion Aλ ⊂ Aµ .

By assumption Aλ ⊂ Aµ for 0 < µ < λ . We deduce that for any λ > 0 , Aλ is

an accretive operator; indeed, for µ > 0 small enough ( µ < λ ), (I + µAλ)−1 is a non-

expansive mapping since it is contained in Jµ . Thus, for 0 < λ < µ , (I + µAλ)−1 is a

single-valued operator in X0 containing (I + µAµ)−1 = Jµ , which is everywhere defined in

X0 ; so (I + µAλ)−1 = (I + µAµ)

−1 and Aλ = Aµ . ⋄

As we have seen in the proof above, there exists a multivalued operator A+ in L1(IRN )∩L∞(IRN ) such that J+

λ = (I + λA+)−1 for any λ > 0 . This operator is accretive densely

defined in L1(IRN) and R(I + λA+) = D(J+λ ) = L1(IRN)∩L∞(IRN) is dense in L1(IRN)

for any λ > 0 . This operator A+ is exactly defined by

v ∈ A+u ⇔ ∃f ∈ L1(IRN) ∩ L∞(IRN) such that

u is the maximum bounded g.e.s. of (E) and v = f − u;

it follows that A+ is actually single-valued since for v ∈ A+u one has v = divxφ(u) in

D′(IRN) . By the Crandall-Liggett theorem (cf. [CL71, B72, C76, BCP]) for any (f, g) ∈L1(IRN)×L1(Q) there exists a unique mild (or integral) solution u ∈ C([0, T ];L1(IRN )) of

du

dt+ A+u = g on (0, T ), u(0) = f. (11)

Theorem 2 With the notations above, for (f, g) satisfying (1) with c = 0 , the mild solution

of (11) is the maximum bounded g.e.s. of (CP).

Proof: With the same argument as in [C72] and [B72], it is clear that, under the assumptions

(1), the mild solution u ∈ C([0, T ];L1(IRN )) of (11) is in L∞(Q) and a g.e.s. of (CP).

Therefore u ≤ u a.e. on Q .

Now we prove that u satisfies∫(u(t)− w)+ ≤

∫(f − w)+ +

∫ t

0

[u(τ)− w, g(τ)− A+w

]+dτ (12)

for a.a. t ∈ (0, T ) and for all w ∈ D(A+) , where for u, f ∈ L1(IRN)

[u, f

]+=

∫

u>0

f +

∫

u=0

f+ ≡ infµ>0

∫(u+ µf)+ −

∫u+

µ.

Using translation invariance in time, we shall have ddt

∫(u(t)−w)+ ≤

[u(t)−w, g(t)−A+w

]+

in D′((0, T )) . Applying the results of [BaB92], we shall conclude that u ≤ u a.e. on Q

and this will end the proof.

Let w ∈ D(A+) , δ > 0 . By definition w = J+1 h , A+w = h− J+

1 h with some

h ∈ L1(IRN)∩L∞(IRN) . Consider wδ a bounded g.e.s. of w+divx φ(w) = h+δ . Take wδ

as a stationary bounded g.e.s. of the corresponding (CP); since wδ − δ ∈ L∞(0, T ;L1(IRN))

and u ∈ L∞(0, T ;L1(IRN)) (cf. Corollary 2.1 in [BK96]), we have λN+1(u > wδ

)<∞

with the same argument as in the proof of Theorem 1. Thus Lemma 1b) yields

∫ (u(t)− wδ

)+≤

≤∫ (

f − wδ)+

+

∫ t

0

[u(τ)− wδ, g(τ)− (h+δ−wδ)

]+dτ ≤

∫ (f − wδ

)++

+

∫ t

0

1

µ

∫ (u(τ)− wδ + µ(g(τ)−A+w+wδ−w−δ)

)+−∫ (

u(τ)− wδ)+

dτ

(13)

for any µ > 0 . As δ decreases to 0 , wδ decreases to w ; moreover for 0 < µ ≤ 1 ,(u(τ)− wδ + µ(g(τ)−A+w+wδ−w−δ)

)+increases to

(u(τ)− w + µ(g(τ)− A+w)

)+.

So we may pass to the limit in (13) and obtain

∫ (u(t)− w

)+≤

≤∫ (

f − w)+

+

∫ t

0

1

µ

∫ (u(τ)− w + µ(g(τ)− A+w)

)+−∫ (

u(τ)− w)+

dτ

for any 0 < µ ≤ 1 . Letting µ → 0 yields (12). ⋄

Remark 1 Of course, one may consider minimum bounded g.e.s. of (10), define the corre-

sponding operator A− , and prove the following result analogous to Theorem 2:

The mild solution of (11) with A− in place of A+ is exactly the minimum bounded

g.e.s. of (CP).

Corollary 1 For a given continuous flux function φ and c ∈ IR , the following assertions are

equivalent:

(i) for all f = f0 + c with f0 ∈ L1(IRN) ∩ L∞(IRN ) there exists a unique bounded

g.e.s. of (E);

(ii) for all (f, g) satisfying (1) there exists a unique bounded g.e.s. of (CP).

1.IV.3. Uniqueness results in L1(IRN) ∩ L∞(IRN) 91

Proof: Replacing φ(r) by φ(r + c) , we may assume c = 0 .

If (i) holds, the operators A+ and A− coincide and then by Theorem 2, (see also

Remark 1), for any (f, g) satisfying (1) the maximum and minimum bounded g.e.s. of (CP)

coincide, so that (ii) holds.

Conversely, assume that (ii) holds and for f ∈ L1(IRN) ∩ L∞(IRN) let u, u be two

bounded g.e.s. of (E). Then u(t) ≡ u is a bounded g.e.s. of (CP) corresponding to (u, g(t)≡f−u) and so, by uniqueness, it is the maximum bounded g.e.s. and then, by Theorem 2,

it is the unique mild solution of the corresponding evolution problem (11). In the same way

u(t) ≡ u is the unique mild solution of (11) corresponding to (u, g(t)≡f−u) . Then by the

integral inequality (see [B72, BCP, BW94])

−∫

|u− u| =[u(t)− u(t), (f − u)− (f − u)

]≥ d

dt

∫|u(t)− u(t)| = 0,

where [·, ·] stands for the bracket associated with the standard norm in L1 , i.e., for all

u, f ∈ L1(IRN ) ,[u, f

]=

∫

u 6=0

f sign u+

∫

u=0

|f | . It follows that u = u a.e. in IRN

so that (i) holds. ⋄

Remark 2 For f = f0 + c with f0 ∈ L1(IRN) ∩ L∞(IRN) , any bounded g.e.s. of (E) is

in c + L1(IRN) (cf. Corollary 2.1 in [BK96]); so there is uniqueness of a bounded g.e.s. to

(E) if and only if

∫ (u(f)− u(f)

)= 0 , where u(f) and u(f) are the maximum and the

minimum bounded g.e.s. of (E), respectively.

By Part 1) of Proposition 2, for given c ∈ IR the map f0 7→ u(f0+c)−c is a contraction

for the L1 -norm; the same holds for f0 7→ u(f0+c)−c so that f0 7→∫ (

u(f0+c)−u(f0+c))

is continuous for the L1 -topology. It follows that (i) of Corollary 1 is equivalent to the

uniqueness of a bounded g.e.s. of (E) for all f0 in some L1 -dense subset of L1(IRN ) ∩L∞(IRN ) .

Consequently, since the L1 -topology in L1(IRN) ∩ L∞(IRN ) is separable, Proposition 1

can be improved as follows.

Proposition 3 There exists an at most countable set N in IR such that, for all c ∈ IR\N ,

the two properties (i) and (ii) of Corollary 1 hold.

3 Some uniqueness results in L1(IRN) ∩ L∞(IRN)

As noted in the introduction, we still do not know if, for any continuous flux function φ , there

is uniqueness of a bounded g.e.s. to (CP) under assumption (1) or to (E) for all f = f0 + c ,

f0 ∈ L1(IRN) ∩ L∞(IRN) , c ∈ IR . In this section we shall improve some uniqueness results

shown in [BK96].


Theorem 3 Assume there exist orthonormal vectors ξ1, . . . , ξN−1 and C : IR → [0,+∞)

continuous such that

d

drξi · φ(r) ≤ C(r) in D′(IR) for i = 1, . . . , N − 1. (14)

Then for any c ∈ IR the two properties of Corollary 1 hold.

We shall need the following lemma.

Lemma 4 Let ξ ∈ IRN , ξ 6= 0 , such that r ∈ IR 7→ ξ · φ(r) is nondecreasing. Let α ∈ IR

and f ∈ L∞(IRN) with support contained in H = x ∈ IRN ; ξ · x ≥ α. Assume that one

of the following conditions holds:

a) there exists a unique bounded g.e.s. of (E);

b) f ∈ L∞0 (IRN) .

Then for all bounded g.e.s. u of (E) the support of u is also contained in H .

Proof: a) This is clearly true if φ is locally Lipschitz continuous. Indeed, by the definition

of g.e.s.,

∫|u(x)|ρ

(α− ξ · x

ε

)ζ(x)dx ≤

∫|f(x)|ρ

(α− ξ · x

ε

)ζ(x)dx+

+

∫sign u(x)(φ(u(x))− φ(0))

−ξερ′(α− ξ · x

ε

)ζ(x) + ρ

(α− ξ · x

ε

)Dζ(x)

dx

for ζ ∈ D(IRN) , ζ ≥ 0 , ρ ∈ C∞(IR) with ρ′ ≥ 0 , ρ = 0 on (−∞, 0] , ρ = 1 on

[1,+∞) , and ε ≥ 0 . Since sign u(x)(φ(u(x)) − φ(0)) · ξ ≥ 0 and f(x)ρ(α−ξ·x

ε

)≡ 0 ,

using the Lipschitz continuity of φ we get

∫|u(x)|ρ

(α− ξ · x

ε

)ζ(x)dx ≤ C

∫|u(x)|ρ

(α− ξ · x

ε

)|Dζ(x)|dx.

Let ε → 0 and ζ → 1 ; it follows that u = 0 a.e. in IRN \H (see, for instance, Lemma

1.1 in [BK96]).

For the general case, let φn = φ ∗ ρn , where ρn is a sequence of mollifiers, and let un

be the bounded g.e.s. of (E) corresponding to the flux φn . Using the contraction property

and translation invariance, we see that the sequence un is relatively compact in L1loc(IR

N) ;

clearly any limit point is a bounded g.e.s. of (E) and then by the uniqueness assumption u is

the limit in L1loc(IR

N) of the sequence un . Note that r ∈ IR 7→ ξ ·φn(r) is nondecreasing

for all n ; thus by the argument above supp un ⊂ H and the same is true at the limit.

b) Let f ∈ L∞0 (IRN) . By Theorem 1 the equation (E) has a maximum bounded g.e.s.

u , which is the limit of any sequence un of bounded g.e.s. of (E) corresponding to

fn = f + cn with cn ↓ 0 . Moreover, by Proposition 1 we may choose cn so that there

is uniqueness of bounded g.e.s. of (E) corresponding to fn . By the first part of Lemma 4,

supp (un − cn) ⊂ H , therefore supp u ⊂ H . Using the same argument for the minimum

bounded g.e.s. u of (E), we see that the conclusion of Lemma 4 still holds. ⋄

Proof of Theorem 3: Replacing φ(r) by φ(r+ c)− φ(c) , we may assume c = 0 and

φ(0) = 0 .

Since we are working with bounded solutions, we may also assume that C(r) is constant.

By replacing ξi by −ξi , it is then equivalent to assume instead of (14) that

d

drξi · φ(r) + C ≥ 0 in D′(IRN ) for i = 1, . . . , N − 1.

Now we notice that for η ∈ IRN , u is a bounded g.e.s. of (CP) corresponding to

φ, f, g if and only if u(t, x) = u(t, x − tη) is a bounded g.e.s. of (CP) corresponding to

φ(r) = φ(r)+ rη , f(x) = f(x) , and g(t, x) = g(t, x− tη) . Then, according to Corollary 1,

the conclusion of Theorem 3 holds for φ if and only if it holds for the flux function φ(r)+rη .

Choosing η ∈ IRN such that η · ξi > C for i = 1, . . . , N −1 , which is always possible since

the vectors are linearly independent, we may assume that

r ∈ IR 7→ ξi · φ(r) ∈ IR is an increasing homeomorphism for i = 1, . . . , N − 1, (15)

a slightly strengthened version of (3).

Under the assumption (15), we prove the result by induction in the dimension N . The

result is true for N = 1 (see [B72]). Assuming that it is true for N − 1 , we prove it for

N ≥ 2 . Changing coordinates, we may assume from (15) that φ(r) =(φ1(r), . . . , φN(r)

)

with φi(·) increasing homeomorphism from IR to IR for i = 1, . . . , N − 1 . We shall

prove that the equation (E) has a unique bounded g.e.s. for any f ∈ L1(IRN) ∩ L∞(IRN) .

According to Corollary 1, this will end the proof of the theorem.

So let f ∈ L1(IRN)∩L∞(IRN) and u be a bounded g.e.s. of (E); one has u ∈ L1(IRN)

(see Corollary 2.1 in [BK96]). Set x = (x1, x′) with x′ = (x2, . . . , xN) , w(x1, x

′) =

φ1(u(x1, x′)) , β = φ−1

1 , ψ(r) =(φ2(β(r)), . . . , φN(β(r))

). Suppose that w(x1 + t, ·) →

w(x1, ·) in L1loc(IR

N−1) as t → 0 for some x1 ∈ IR ; then for every T > 0 the function

v : (t, x′) ∈ Q′ = (0, T )× IRN−1 7→ w(x1 + t, x′) is a bounded g.e.s. of the Cauchy problem

∂v

∂t+ divx′ ψ(v) = g on Q′, v(0, ·) = v0(·) on IRN−1, (16)

where v0(x′) = w(x1, x

′) and g(t, x′) = f(x1+t, x′)−β(w(x1+t, x′)) ; g ∈ L1(Q′)∩L∞(Q′)

since f and β(w) = u are in L1(IRN ) ∩ L∞(IRN) .

According to Remark 2, it suffices to prove the uniqueness of a bounded g.e.s. of (E)

corresponding to compactly supported f . So assume supp f ⊂ H = x1 ≥ α0 and suppose

there exist u, u two bounded g.e.s. of (E). By Lemma 4, supp u ⊂ H , supp u ⊂ H . Take

x1 = α < α0 ; consider v(t, x′) = φ1(u(t+α, x

′)) , v(t, x′) = φ1(u(t+α, x′)) . The functions

v, v are bounded g.e.s. of (16) corresponding to(v0(·) ≡ 0 , g(t, ·) = f(t+α, ·)−β(v(t, ·))

)

and(v0(·) ≡ 0 , g(t, ·) = f(t+ α, ·)− β(v(t, ·))

), respectively.

By the inductive assumption the Cauchy problem (16) has a unique bounded g.e.s., which

is in L1(IRN−1) for a.a. t ∈ (0, T ) ; as in the proof of Corollary 1, it follows that the integral

inequality holds:

d

dt

∫|v(t)− v(t)| ≤

[v(t)− v(t), g(t)− g(t)

]= −

∫|β(v(t))− β(v(t))| ≤ 0.

Hence

∫|v(t)− v(t)| ≤

∫|v0 − v0| = 0 , so that v = v a.e. in Q′ . Thus u = u a.e. in

H , which proves the Theorem. ⋄In [B72] it has been proved that for any f ∈ L1(IRN) ∩ L∞(IRN ) there is uniqueness of

u ∈ L1(IRN)∩L∞(IRN) g.e.s. of (E) under the isotropic assumption limr→0

‖φ(r)‖

r1−1/N = 0 . In the

next theorem we shall prove the uniqueness under the anisotropic assumption introduced in

[KP90] and [BK96].

Theorem 4 Let c ∈ IR and ω1, . . . , ωN be moduli of continuity, i.e., increasing sub-additive

continuous functions from [0, δ] into [0,+∞) , δ > 0 , with ωi(0) = 0 , i = 1, . . . , N ;

assume that

lim infr→0

1

rN−1

N∏

i=1

ωi(r) <∞. (17)

Assume that there exist orthonormal vectors ξ1, . . . , ξN such that |ξi·φ(c+r)−ξi·φ(c)| ≤ωi(|r|) for all r ∈ [−δ, δ] , i = 1, . . . , N . Then the two assertions of Corollary 1 hold†.

Proof: We may assume that c = 0 , φ(0) = 0 and φ = (φ1, . . . , φN) with |φi(r)| ≤ωi(|r|) for r ∈ [−δ, δ] , i = 1, . . . , N . Recalling Remark 2 and Corollary 1, we only need

to prove for f, u ∈ L1(IRN) ∩ L∞(IRN ) with u g.e.s. of (E) that

∫u =

∫f . Replacing

†In spite of the fact that this result underlines one more time (cf. [B72]) that Holder continuity of the flux

function at zero simplifies the issue of uniqueness in L1(IRN )∩L∞(IRN ) , Proposition 3 hereabove suggests

that the non-uniqueness, if there is any, can have no intrinsic relation with regularity properties of the flux

function. As we see, the appropriate Holder continuity of the flux function permits to pass to the limit directly

in the estimations of Lemma 3.1 in [BK96], and thus obtain the contraction property. But even in the case

one seems unable to deduce this property, there is uniqueness for a.a. translations φ(· − c)−φ(·) of the flux

function φ(·) , with c ∈ IR . This should be compared to the fact that one can construct functions that have

equally bad Holder continuity properties at all points of their domain; for exemple, one can use realisations of

a Wiener process. Thus we can suggest that, if the uniqueness in L1(IRN )∩L∞(IRN ) is determined by some

regularity property of the flux function, this must be a property that holds on the domain of any continuous

function φ(·) everywhere, except an at most countable set of points. The author ignores what could be the

nature of such property.

On the other hand, note that the assumption of Theorem 3 makes appeal to a global property of the flux

function, which also differs from the “pointwise regularity” point of view suggested by Theorem 4.

ωi(r) , φi , f , and u by ωi(Mr)/M , φ(Mr)/M , f/M , and u/M , respectively, we

may assume that ‖u‖∞ ≤ δ .

Clearly, it suffices to show that for all µ > 0, R > 0 there exists a function ζ such that

0 ≤ ζ ≤ 1 on IRN , ζ(x) = 1 for all x ∈ [−R,R]N , and

∣∣∣∣∫

(u− f)ζ

∣∣∣∣ < µ; (18)

for this we follow the proof of Lemma 1.1 in [BK96].

For r > 0 set λi(r) = ωi(r)/r . If all λi are bounded, then φ is Lipschitz continuous

and the result is well known (see the Introduction). Without loss of generality we may assume

that limr→0

λi(r) = +∞ for i = 1, . . . , l and λi(r) ≤ λ for i = l+1, . . . , N with some

l ∈ 1, . . . , N . Since ωi are sub-additive and positive for r > 0 , ωi(r) ≥ λ0r for some

λ0 > 0 , so that it is equivalent to assume instead of (17) that lim infr→0

C(r) = C <∞ , where

C(r) = rλN−ll∏

i=1

λi(r) . Note that if l = 1 , then clearly C = 0 .

For all u bounded g.e.s. of (E), for all ζ ∈ D(IRN ) we have

∫uζ =

∫φ(u) ·Dζ +

∫fζ. (19)

Moreover, since f and u are bounded, (19) is also valid for ζ given by

ζ(x1, . . . , xN ) =N∏

i=1

exp

(−( |xi|Ri

− 1)+)

with arbitrary positive Ri . Take ζ corresponding to Ri = λi(ε)/η for i = 1, . . . , l and

Ri = λ/α for i = l+1, . . . , N ; positive numbers α, η, ε will be chosen later. We note

that 0 ≤ ζ ≤ 1 , ζ(x) ≡ 1 onN∏i=1

[−Ri, Ri] ,

∫ζ =

22N

ηlαN−lλN−l

l∏

i=1

λi(ε) , and |Diζ | =

ηζ

λi(ε)χ|xi|>Ri for i = 1, . . . , l , |Diζ | =

αζ

λχ|xi|>Ri for i= l+1, . . . , N .

From (19) we get

∣∣∣∣∫(u− f)ζ

∣∣∣∣ ≤N∑

i=1

∫ωi(|u|)|Diζ | .

Now, by the sub-additivity of ωi , for i = 1, . . . , l we have ωi(r) ≤ rωi(ε)/ε+ ωi(ε) =

rλi(ε)+ελi(ε) for all ε > 0 ; for i= l+1, . . . , N we have ωi(r) ≤ rλ . Hence by substituting

into the last estimate the expressions above for |Diζ | and

∫ζ , we get

∣∣∣∣∫

(u− f)ζ

∣∣∣∣ ≤l∑

i=1

η

∫

|xi|>Ri

|u|ζ +l∑

i=1

εη

∫

|xi|>Ri

ζ +N∑

i=l+1

α

∫

|xi|>Ri

|u|ζ ≤

≤ ηl∑

i=1

∫

|xi|>Ri

|u|+ l22N

ηl−1αN−l· ελN−l

l∏

i=1

λi(ε) + α(N − l)

∫|u|.

Take µ > 0 , R > 0 . Choose α0 > 0 such that λ/α0 > R and α0 ·(N−l)‖u‖1 < µ/3 .

Choose η0 > 0 such that1

ηl−10

· l22N

αN−l0

· 2C < µ/6 ; note that if l = 1 then C = 0

and whatever η0 is good. Finally, since λi(ε) → ∞ as ε → 0 for i = 1, . . . , l and

u ∈ L1(IRN) , by definition of C there exists ε0 > 0 satisfying ε0λN−l

l∏

i=1

λi(ε0) < 2C +

µ

6· α

N−l0 ηl−1

0

l22Nsuch that Ri = λi(ε0)/η0 > R ,

l∑

i=1

∫

|xi|>Ri

|u| < µ

3η0. It follows that (18)

holds for ζ constructed with α0, η0, ε0 . ⋄

Remark 3 Introducing ξ1, . . . , ξN in the condition (17) is not superfluous. Indeed, take

N = 2 and let φ =(u , u/|u|2/3

)in some orthonormal basis ξ1, ξ2 ; here (17) holds.

Changing coordinates by rotation by any angle θ such that θ 6= πk/2, k ∈ ZZ , we see that

condition (17) fails in the new basis.

Appendix: proof of Lemma 2

We give here the complete proof of Lemma 2. More precisely, we shall prove the following

result:

Theorem 5 Let φ : IRN 7→ IR be a continuous function.

a) There exists a map G : L∞(IRN) 7→ L∞(IRN) satisfying:

(i) for any f ∈ L∞(IRN), u = Gf is a g.e.s. of (E);

(ii) G is a T-contraction for the L1-norm, i.e., for any f, f ∈ L∞(IRN)

∫ (Gf −Gf

)+≤∫ (

f − f)+.

b) Set X =(f, g) ; (f, g) satisfies (2)

; there exists a map U : X 7→ L∞(Q)∩

C([0, T ];L1loc(IR

N)) satisfying:

(i) for any (f, g) ∈ X , u = U(f, g) is a g.e.s. of (CP);

(ii) for any (f, g), (f , g) ∈ X , the T-contraction property holds:

supt∈[0,T ]

∫

IRN

(U(f, g)(t)− U(f , g)(t)

)+≤∫

IRN

(f − f

)++

∫∫

Q

(g − g

)+.

Proof of a): First, let f ∈ L1(IRN) ∩ L∞(IRN ) . For ε > 0 , take φε : IR → IRN

Lipschitz continuous functions such that φε converge to φ uniformly on compact sets in

IR , as ε→ 0 . It is well-known that there exists a unique solution uε to the equation

uε + divx φε(uε) = ε∆x u

ε + f on IRN ;

moreover, the map Gε : f ∈ L1(IRN) ∩ L∞(IRN) 7→ uε ∈ L1(IRN ) ∩ L∞(IRN) is a T-

contraction for the L1 -norm, the maximum principle ( ‖uε‖∞ ≤ ‖f‖∞ ) holds and there is

1.IV.4. Appendix 97

translation invariance in x . Thus the family Gεfε>0 is relatively compact in L1loc(IR

N) .

Take a countable L1 -dense set M in L1(IRN) ∩ L∞(IRN) ; by the diagonal process, there

exist εn → 0 such that Gεnf → u =: G0f in L1loc(IR

N) for all f ∈ M . It is clear

that u is a g.e.s. of (E) and G0 : M 7→ L1(IRN) ∩ L∞(IRN) is a T-contraction for the

L1 -norm. Thus G0 can be extended to the whole of L1(IRN) ∩ L∞(IRN) so that G0 is a

T-contraction for the L1 -norm, G0f is a g.e.s. of (E) and the maximum principle holds.

Now for the general case f ∈ L∞(IRN) , set fn,m = f+χ|x|≤n−f−χ|x|≤m ∈ L1(IRN)∩L∞(IRN ) . As n → ∞ , G0fn,m ↑ um ∈ L∞(IRN) ; further, as m → ∞ , um ↓ u =:

Gf . It is clear that u is a bounded g.e.s. of (E); by the Fatou Lemma, it follows that∫ (Gf − Gf

)+≤ lim inf

n→∞,m→∞

∫ (fn,m − fn,m

)+for f, f ∈ L∞(IRN ) . It is easy to check

that limm→∞

limn→∞

∫ (fn,m − fn,m

)+=

∫ (f − f

)+∈ [0,+∞] , so that (ii) also holds. ⋄

Proof of b): First, set X0 =[L1(IRN)∩L∞(IRN)

]×[L1(Q)∩L∞(Q)

]and let (f, g) ∈

X0 . For ε > 0 , take φε as in the proof of a); there exists a unique solution uε to the

Cauchy problem

∂uε

∂t+ divx φ

ε(uε) = ε∆x uε + g on Q

uε(0, ·) = f on IRN ;

moreover, the map Uε : (f, g) ∈ X0 7→ uε ∈ L1(Q) ∩ L∞(Q) ∩ C([0, T ];L1loc(IR

N)) satisfies

the maximum principle ( ‖uε‖∞ ≤ ‖f‖∞ +

∫ T

0

‖g(τ, ·)‖∞dτ ), the T-contraction property

holds , and there is translation invariance in x . Hence there exists a modulus of continuity

ωf,g such that∫ ∣∣∣Uε(f, g)(t, x+∆x)− Uε(f, g)(t, x)

∣∣∣dx ≤ ωf,g(∆x)

uniformly in ε > 0 and t ∈ [0, T ] . By Theorem 2 in [K69a], it follows that for any compact

set K ⊂ IRN

∫

K

∣∣∣Uε(f, g)(t+∆t, x)− Uε(f, g)(t, x)∣∣∣dx ≤ ωf,g,K(∆t)

uniformly in ε > 0 and t ∈ [0, T ] , where ωf,g,K is a modulus of continuity. Take a

countable set M dense in X0 for the L1(IRN)×L1(Q) -topology. By the diagonal process,

there exist εn → 0 such that Uεn(f, g) → u =: U0(f, g) in L1loc(Q) for all (f, g) ∈ M ;

u is a g.e.s. of (CP), and the maximum principle and the T-contraction property hold for

U0 : M 7→ L1(Q) ∩ L∞(Q) ∩ C([0, T ];L1loc(IR

N)) . Thus U0 can be extended to the whole

of X0 , so that U0(f, g) is a g.e.s. of (CP), the T-contraction property holds and there is

translation invariance in x and the maximum principle holds.

Now for the general case (f, g) ∈ X , set fn,m = f+χ|x|≤n − f−χ|x|≤m and gn,m =

minn, g+χ|x|≤n−minm, g−χ|x|≤m , so that we have (fn,m, gn,m) ∈ X0 . As n→ ∞ ,


U0(fn,m, gn,m) ↑ um ∈ L∞(Q) ; further, as m → ∞ , um ↓ u =: U(f, g) ∈ L∞(Q) . By the

Fatou Lemma, it follows that (ii) holds.

We now show that u = U(f, g) is in C([0, T ];L1loc(IR

N )) and a g.e.s. of (CP). Indeed,

there exists an increasing sequence (ni, mi)i∈IN in IN2 such that ui = U0(fi, gi) → u

in L1loc(Q) as i → ∞ , where fi = fni,mi

, gi = gni,mi. By Lemma 3.1 in [BK96] and

translation invariance, for ζ ∈ D(IRN )

supt∈[0,T ]

∫ ∣∣∣ui(x+∆x, t)−ui(x, t)∣∣∣ ζ(x) dx ≤

∫ ∣∣∣fi(x+∆x)−fi(x)∣∣∣ ζ(x) dx+

+

∫ T

0

∫ ∣∣∣gi(s, x+∆x)−gi(s, x)∣∣∣ ζ(x) dxds+

∫ T

0

∫ ∣∣∣φ(ui(s, x+∆x))−φ(ui(s, x))∣∣∣∣∣∣Dζ(x)

∣∣∣ dxds.

The last term tends to

∫ T

0

∫ ∣∣∣φ(u(s, x+∆x))−φ(u(s, x))∣∣∣∣∣∣Dζ(x)

∣∣∣ dxds as i→ ∞ , therefore

for any compact set K ⊂ IRN

∫

K

∣∣∣ui(t, x+∆x)− ui(t, x)∣∣∣dx ≤ ωf,g,K(∆x)

uniformly in i ∈ IN and t ∈ [0, T ] , where ωf,g,K is a modulus of continuity. Hence, again by

Theorem 2 in [K69a], the family ui(t, ·)i∈IN is equicontinuous from [0, T ] to L1loc(IR

N) .

Thus u ∈ C([0, T ];L1loc(IR

N)) and u(0, ·) = limi→∞

fi = f , so that u is a bounded g.e.s. of

(CP). ⋄

Part 2

Weak Solutions

for Elliptic-Parabolic Systems

CHAPTER 2.I

Elliptic-Parabolic Problems: Existence

and Continuity with Respect to the Data

of Weak Solutions†

Introduction

Let Ω be a bounded domain of IRd with Lipschitz boundary ∂Ω . For i = 1, . . . , N , let

Γ0,i be a closed set in ∂Ω and Γ1,i = ∂Ω\Γ0,i . We consider initial boundary value problems

for elliptic-parabolic systems of the form

b(·, v)t = div a(·, v, Dv) + f(·, v) on Q = (0, T )× Ω

ai(·, v, Dv) · ν = gi(·, v) on Σ1,i = (0, T )× Γ1,i for i = 1, . . . , N

vi = hi on Σ0,i = (0, T )× Γ0,i for i = 1, . . . , N

b(·, v)(0, ·) = u0 on Ω,

(P)

where b : Q× IRN 7→ IRN , a = (a1, . . . , aN) : Q× IRN × (IRd)N 7→ (IRd)N , f : Q× IRN 7→IRN , g = (g1, . . . , gN) : Σ × IRN 7→ IRN , h = (h1, . . . , hN) : Σ 7→ IRN , u0 : Ω 7→ IRN ,

and ν is the external unit normal vector to Σ = (0, T )× ∂Ω . We denote by “ · ” the scalar

product in IRN , and by “ : ” the scalar product in (IRd)N . Except in Section 3.3, we assume

that |Γ0| =∏N

i=1 |Γ0,i| > 0 .

The assumptions on the data will be made precise in Section 1. In the particular case

b(t, x, z) = b(z) , a(t, x, z, ξ) = a(z, ξ) , f ≡ 0 , g ≡ 0 , h ≡ 0 , the assumptions reduce to

the three following conditions on b, a, u0 :

b = ∂Φ, with Φ : IRN 7→ IR convex differentiable ,Φ(0) = 0; (1)

†This chapter is being prepared upon publication [BAB]

102 Continuous Dependence for Elliptic-Parabolic Problems

as usual, we set B(z) =

∫ 1

0

(b(z) − b(σz)

)· z dσ = b(z) · z − Φ(z) ≥ 0 for any z ∈ IRN ;

a is continuous, monotone in ξ, and satisfies Leray-Lions type conditions :

there exist c > 0, C ≥ 0 and a “sublinear” function L : IR+ 7→ IR+

(i.e., a function with limr→∞

L(r)/r = 0) such that

a(z, ξ) : ξ ≥ c|ξ|p −L(|z|p)− C(1 +B(z)

),

|a(z, ξ)|p′ ≤ L(B(z) + |z|p

)+ C

(1 + |ξ|p

),

(2)

with 1 < p <∞ , p′ = p/(p− 1) ;

u0 ∈ L1(Ω; IRN ) with Ψ(u0) ∈ L1(Ω), (3)

where Ψ is the Legendre conjugate of Φ defined by Ψ(z∗) = supz∈ IRN

(z · z∗ − Φ(z)

)∈

[0,+∞] for any z∗ ∈ IRN . Note that one has B(z) = Ψ(b(z)) for all z ∈ IRN . In this

chapter, adding the structure conditiona(z, ξ) = a(b(z), ξ) for all z ∈ IRN , ξ ∈ (IRd)N ,

where a : R(b(·))× (IRd)N 7→ (IRd)N is Caratheodory,(4)

we prove existence of a weak solution (in the variational sense) to (P ) .

Such result has already been proved by Alt-Luckhaus in [AL83], while under more restrictive

assumptions. Since then, many similar results have been obtained (e.g., cf. [Kac90, DiDT94,

FKac95, BW96, Bou97]), in particular in the case of time-space dependent elliptic part and

space dependent parabolic part. Equations with the parabolic part of the form c(t, x, v) vt

have been studied in [Pl98, Pl00] by a different approach; existence results for some of equa-

tions of this form can be derived from the ours, upon introducing b(t, x, z) =

∫ z

0

c(t, x, ζ) dζ .

The aim and the interest of this work is, first, in proving the continuous dependence of

weak solutions on the data in (P ) ; existence is an easy corollary of this result. Secondly,

we use arguments that permit to make less restrictive assumptions (in particular, all data are

(t, x) - dependent) and, at the same time, clarify the essence of the proof.

The continuity theorem is presented in Section 1 (Theorem 1); in the reduced case con-

sidered above it reads as follows:

Theorem 0 Let(bn, an, u0,n

)be a sequence of data satisfying the assumptions (1)-(3) with

c, C,L(·) independent of n ∈ IN and corresponding Bn,Ψn . Assume that bn(·) → b(·) in

C(IRN ; IRN ) †, for all ξ ∈ (IRd)N an(·, ξ) → a(·, ξ) in C(IRN ; (IRd)N) , and u0,n → u0 a.e

on Ω with Ψn(u0,n) → Ψ(u0) in L1(Ω) . For n = 1, 2, . . . let vn be a weak solution of

the corresponding problem (Pn) with f ≡ 0 , g ≡ 0 , h ≡ 0 , and where |Γ0| > 0 . Then

†I.e., for all compact subset K of IRN there is uniform convergence for z ∈ K

2.I.1. Assumptions and results 103

(i) The sequence vn is bounded in Lp(0, T ;W 1,p(Ω; IRN)) and the sequence bn(vn)is relatively compact in L1(Q; IRN) .

(ii) Any weak limit point of vn in Lp(0, T ;W 1,p(Ω; IRN)) is a weak solution of (P ) ,

provided (P ) satisfies (4).

A precise definition of a weak solution is given in Section 1. Note that, while not assuming

the strict monotonicity of b in z , we have to impose the structure condition (4) , which

is trivial for b strictly monotone. However, this condition is not intrinsically related to the

existence of weak solution to (P ) (cf. [BW99] and Section 3.2).

The proof of the continuity theorem in general setting is given in Section 2 and includes

three essential arguments. First, we establish a time-dependent version of the chain rule

Lemma 1.5 from [AL83] and apply it to get a priori estimates which imply compactness in

x of vn . Secondly, using a general compactness lemma due to S.N.Kruzhkov ([K69a]) we

deduce the compactness of bn(vn) in (t, x) . Finally, under additional structure conditions,

the Minty-Browder argument is used for passage to the limit in the equation (Pn) . The proofs

of the appropriate versions of this three results (Lemma 1, Lemma 6 and Lemma 7) are given

in the Appendix.

In Section 3 we give some remarks and further existence results. More specifically, we

treat the case of inhomogeneous Dirichlet data, discuss structure conditions of type (4), give

one direct extension to the case |Γ0| = 0 , and indicate possible applications to proving

convergence of approximate methods.

1 Assumptions and results

In this section and Section 2 we state the results for the problem (P ) with h ≡ 0 and

|Γ0| > 0 . See Section 3 for some comments on the general case.

Let Ω be a bounded set in IRd with Lipschitz boundary ∂Ω . For i = 1, . . . , N , let

Γ0,i be a closed set in ∂Ω and Γ1,i = ∂Ω\Γ0,i . Let 1 0 , and set

Q = (0, T )× Ω , Σ = (0, T )× ∂Ω .

Our assumptions include a collection of hypotheses that impose restrictions on the growth

of the coefficients ( (H2), (H3), (H4), (H6), (H7), (H8), (H13) below). We will say that


this collection of hypothesis is satisfied if there exist

c = const > 0,

C = const ≥ 0,

K0 : [0, T ]× Ω× IR+ 7→ IR+ such that

K0(·, r) ∈ L1(Q) and K0(0, ·, r) ∈ L1(Ω) for all r ∈ IR+,

K1 : (0, T ) 7→ IR+, K1 ∈ L1(0, T ),

K2 : Q 7→ IR+, K2 ∈ L1(Q),

K3 : Σ 7→ IR+, K3 ∈ L1(Σ),

(5)

for all ε > 0 there exist Kε , Kε1 , K

ε2 and Kε

3 such that

Kε = const ≥ 0,

Kε1 : (0, T ) 7→ IR+, Kε

1 ∈ L1(0, T ),

Kε2 : Q 7→ IR+, Kε

2 ∈ L1(Q),

Kε3 : Σ 7→ IR+, Kε

3 ∈ L1(Σ),

(6)

and if the corresponding inequalities holds.

Let b : [0, T ]× Ω× IRN 7→ IRN satisfy

b(t, x, z) = ∂Φ(t, x, ·)(z), where Φ : Q× IRN 7→ IR

is convex differentiable in z and Φ(t, x, 0) = 0 for all t ∈ [0, T ], x ∈ Ω;(H1)

|b(t, x, z)| ≤ K0(t, x, |z|) for all z ∈ IRN , t ∈ [0, T ] and a.a. x ∈ Ω. (H2)

Set B(t, x, z) =

∫ 1

0

(b(t, x, z) − b(t, x, σz)

)· z dσ , and Ψ(t, x, z∗) = Φ(t, x, ·)∗(z∗) =

supz∈ IRN

(z ·z∗−Φ(t, x, z)

)for any z∗ ∈ IRN . One has B(t, x, z) = b(t, x, z) ·z−Φ(t, x, z) =

Ψ(t, x, b(t, x, z)

).

Note that one has (e.g., cf. Remark 1.2 in [AL83]) for all δ > 0 ,

|b(t, x, z)| ≤ δB(t, x, z) + sup|ζ|≤1/δ

|b(t, x, δ)| ≤ δB(t, x, z) +K0(t, x, 1/δ). (7)

In addition to (H1), (H2) we assume that Φ is absolutely continuous in t on [0, T ]

for all z ∈ IRN and a.a. x ∈ Ω , and that there exists Φt : Q × IRN 7→ IR Caratheodory

such thatd

dtΦ(·, x, z)(t) = Φt(t, x, z) for all z ∈ IRN and a.a. (t, x) ∈ Q . Moreover, we

require that for a.a. x ∈ Ω , a.a. t, s ∈ (0, T ) and all z ∈ IRN the function Φt satisfy

|Φt(t, x, z)| ≤ ε(K1(t)B(s, x, z) + |z|p

)+Kε

2(t, x). (H3)


It follows that Φ, b, B are measurable in x for all t ∈ [0, T ] and z ∈ IRN , and continuous

in (t, z) for a.a. x ∈ Ω .

Let f : Q× IRN 7→ IRN be Caratheodory and assume for a.a. (t, x) ∈ Q

|f(t, x, z)|p′ ≤ ε(K1(t)B(t, x, z) + |z|p

)+Kε

2(t, x). (H4)

Let a : Q× (IRN × (IRd)N) 7→ (IRd)N be Caratheodory and assume for a.a. (t, x) ∈ Q(a(t, x, z, ξ)−a(t, x, z, ξ)

): (ξ− ξ) ≥ 0 for all z ∈ IRN , ξ, ξ ∈ (IRd)N , (H5)

a(t, x, z, ξ) : ξ − f(t, x, z) · z + Φt(t, x, z) ≥ c|ξ|p − ε |z|p−−Kε

1(t)B(t, x, z)−Kε2(t, x),

(H6)

|a(t, x, z, ξ)|p′ ≤ ε(K1(t)B(t, x, z) + |z|p

)+Kε

2(t, x) + C|ξ|p. (H7)

Let gi : (0, T )×Γ1,i× IRN 7→ IRN be Caratheodory, i = 1, . . . , N . For convenience, for

all z ∈ IRN extend gi(·, z) by zero on (0, T )× Γ0,i , and assume for a.a. (t, x) ∈ Σ

|g(t, x, z)|p′ ≤ C|z|p +K3(t, x)

g(t, x, z) · z ≤ ε |z|p +Kε3(t, x).

(H8)

Remark 1 The growth restrictions in (H3), (H4), (H6)−(H8) are in fact of the same type

as those in (2). It is convenient to pass to the form chosen above, rather than majorate

|Φt(t, x, z)| , |f(t, x, z)|p′ , etc., by terms of the form L(t, x, B(t, x, z)) and L(t, x, |z|p) ,with “sublinear” functions L(t, x, ·) subject to additional restrictions on their dependence on

(t, x) .

A typical situation where, for instance, the hypothesis (H4) is satisfied, is when one has

|f(t, x, z)|p′ ≤ M(t, x)((B(t, x, z))1/κ + |z|p/κ

)+K2(t, x)

with κ > 1 , M ∈ Lκ′(Q) and K2 ∈ L1(Q) . An assumption of slightly different kind that

is also covered by (H4) is that

|f(t, x, z)|p′ ≤ N (t)L(B(t, x, z)) + L(|z|p) +K2(t, x)

with N ∈ L1(0, T ) and L(·) “sublinear”, independent of (t, x) . Other growth hypotheses

can be simplified in similar ways.

To shorten the notation, for all v : Q 7→ IRN we will denote the function b(t, x, v(t, x))

by b(v) , the function a(t, x, v(t, x), Dw(t, x)) by a(v,Dw) , the function Ψ(τ, x, w(x))

by Ψ(τ, w) and so on.

†It seems that a more natural condition would be (H3) written only for s = t ; see Remark 6 in the

Appendix for a discussion of this issue

Definition 1 For u0 ∈ L1(Ω; IRN) and h ∈ Lp(0, T ;W 1,p(Ω; IRN )) , a weak solution of

(P ) is a function v : Q 7→ IRN satisfying

(i) v ∈ h+ Lp(0, T ;V ) , B(v) ∈ L∞(0, T ;L1(Ω)) (whence b(v) ∈ L1(Q) by (7));

(ii) for all ζ ∈ Lp(0, T ;V ) with ζt ∈ L∞(Q) and ζ(T ) = 0 ,

∫∫

Q

b(v) · ζt +∫

Ω

u0(·) · ζ(0, ·) =∫∫

Q

a(v,Dv) : Dζ −∫∫

Q

f(v) · ζ −∫∫

Σ

g(v) · ζ.

Let denote by V ′ the dual space of V , and by < ·, · > the duality pairing between V ′ and

V . We remark that, according to (i) and the conditions (H4), (H7), (H8) , the condition

(ii) in the Definition 1 makes sense and can be rewritten, as in [AL83], under the equivalent

form:

there exists χ ∈ Lp′(0, T ;V ′) such that

(ii1)

∫∫

Q

b(v) · ζt +∫

Ω

u0(·) · ζ(0, ·) =∫ T

0

< χ, ζ >

for all ζ ∈ Lp(0, T ;V ) with ζt ∈ L∞(Q) and ζ(T ) = 0,

and

(ii2)

∫ T

0

< χ, ζ >=

∫∫

Q

a(v,Dv) : Dζ −∫∫

Q

f(v) · ζ −∫∫

Σ

g(v) · ζ

for all ζ ∈ Lp(0, T ;V ).

According to (ii1) , the distribution derivative of b(v) with respect to t , b(v)t = −χ , is in

Lp′(0, T ;V ′)) .

Following [AL83], in addition to u0 ∈ L1(Ω; IRN) we assume

Ψ(0, u0) ∈ L1(Ω). (H9)

One has the following result.

Lemma 1 Let b satisfy (H1), (H2), (H3) with corresponding functions B,Φ . Assume

that u0 ∈ L1(Ω; IRN) with Ψ(0, u0) ∈ L1(Ω) and v ∈ Lp(0, T ;V

)with B(v) ∈

L∞(0, T ;L1(Ω)) (whence b(v) ∈ L1(Q) ) be given such that (ii1) holds with some χ ∈Lp′(0, T ;V ′) . Then one has Φt(v) ∈ L1(Q) , and for a.a. t ∈ (0, T )

∫

Ω

B(v)(t) =

∫

Ω

Ψ(0, u0)−∫ t

0

< χ(τ), v(τ) > dτ −∫ t

0

∫

Ω

Φt(v)(τ) dτ. (8)

This is a time-dependent version of the chain rule lemma (cf. [Bam77, AL83, Ot96, CaW99]),

which is crucial in this framework. We give a proof of it in the Appendix.


Remark 2 Under the hypothesis we take, we avoid much unnecessary technicalities by re-

quiring into Definition 1 and Lemma 1 that B(v) ∈ L∞(0, T ;L1(Ω)) (which was one of the

claims in the corresponding lemma in [AL83]). Still this property, together with (ii1) and

(ii2) , could be deduced from the assumption that b(v) ∈ L1(Q) and Definition 1 (ii) . We

also take h ≡ 0 in the statement of Lemma 1, hoping to get the general case by translation

by h (cf. Section 3.1).

In order to state the main result of this chapter, we consider a sequence of problems (Pn) ,

n ∈ IN , with data(bn, an, fn, gn, u0,n

)satisfying

(bn, an, fn, gn

)verify (H1)− (H8) with c,K0, K1, K

ε, Kε2 , K3

independent of n and corresponding functions Bn,Φn,t,Ψn;

Ψn(0, u0,n) is bounded in L1(Ω).

(H)

We will say that a sequence of problems (Pn) converge to the problem (P ) with data(b, a, f, g, u0

), if:

(bn(t, x, ·),Φn,t(t, x, ·), an(t, x, ·, ξ), fn(t, x, ·)

)−→

−→(b(t, x, ·),Φt(t, x, ·), a(t, x, ·, ξ), f(t, x, ·)

)

in C(IRN ; IRN × (IRd)N × IRN

)for a.a. (t, x) ∈ Q, for all ξ ∈ (IRd)N ;

gn(t, x, ·) → g(t, x, ·) in C(IRN ; IRN) for a.a. (t, x) ∈ Σ;

u0,n → u0 a.e. on Ω with Ψn(0, u0,n) → Ψ(0, u0) in L1(Ω)

(9)

Theorem 1 Let |Γ0| > 0 and the Dirichlet part of the boundary data be homogeneous, i.e.,

h ≡ 0, (H10)

and(bn, an, fn, gn, u0,n

)be a sequence of data satisfying (H) . Assume that (Pn) converge

to (P) in the sense (9). For n = 1, 2, . . . let vn be a weak solution of the corresponding

problem (Pn) . Then

(i) The sequence vn is bounded in Lp(0, T ;W 1,p(Ω; IRN)) and the sequence bn(vn)is relatively compact in L1(Q; IRN) .

(ii) Any weak limit point of vn in Lp(0, T ;W 1,p(Ω; IRN)) is a weak solution of (P ) ,

provided (P ) satisfies the structure conditions (H11)− (H13) †:f(t, x, z) = f(t, x, b(t, x, z)) for all z ∈ IRN and a.a. (t, x) ∈ Q,

where f : (t, x, β) ∈ Q× IRN | β ∈ R(b(t, x, ·)) 7→ IRN is Caratheodory;(H11)

†By R(b(t, x, ·)) we denote the image of IRN by b(t, x, ·)


a(t, x, z, ξ) = a(t, x, b(t, x, z), ξ) for all z ∈ IRN , ξ ∈ (IRd)N and a.a. (t, x) ∈ Q,

where a : (t, x, β, ξ) ∈ Q× (IRN × (IRd)N) | β ∈ R(b(t, x, ·)) 7→ (IRd)N

is Caratheodory;

(H12)

Φt(t, x, z) = φ(t, x, b(t, x, z)) + z · ϕ(t, x, b(t, x, z)) for all z ∈ IRNand a.a. (t, x) ∈ Q,

where (φ, ϕ) : (t, x, β) ∈ Q× IRN | β ∈ R(b(t, x, ·)) 7→ IR× IRN

is Caratheodory and‡ |ϕ(t, x, b(t, x, z))|p′ ≤ K1(t)Ψ(t, x, b(t, x, z)) + C|z|p +K2(t, x)

for a.a. (t, x) ∈ Q and all z ∈ IRN .

(H13)

Remark 3 In the convergence part, we will manage with the dependence of coefficients on

(t, x) by considering them as mappings from Q to C(IRN ; IRN ) or C(IRN×(IRd)N ; (IRd)N)

and repeatedly applying the Egorov and Lusin theorems. It is possible, since a Caratheodory

function Y1× Y2 7→ Y3 , where Yi ⊂ IRDi , i = 1, 2, 3 , is measurable considered as mapping

from Y1 to C(Y2; Y3) . Indeed, it is weakly measurable since it is measurable in y1 ∈ Y1 for

all fixed y2 ∈ Y2 ; hence it is strongly measurable (e.g., cf. [BbkiXIII],Chap.IV,S5,Prop.10).

Corollary 1 Let the data in (P ) satisfy (H1)− (H13) , and Ψ(u0) ∈ L1(Q) . Then there

exists a weak solution to (P ) with h = 0 .

Proof of Corollary 1: Let approximate (P ) in the sense (H) ,(9) by a sequence of

problems (Pn) with bn bilipschitz in z , and with data and coefficients regular in (t, x) .

Note that we have to assure that Ψn(0, u0,n) → Ψ(0, u0) in L1(Ω) . This is done by

choosing for all n ∈ IN some m ∈ IN and a measurable function u0,n such that u0,n

is a value within the image of [−m,m]N by b(0, x, ·) , and ‖Ψn(0, u0,n) − Ψ(0, u0)‖ ≤1/n . In turn, this is possible since Bn(0, x, ·) are dominated on [−m,m]N by the function√nmK0(0, x,

√nm) ∈ L1(Ω) uniformly in n , since the assumption (H2) holds uniformly

in n .

To prove the existence for (Pn) itself, we use the Galerkin approximations in the way it is

done in [LJLL65, JLL] and [ALpr]. The global in time existence of Galerkin approximations for

(Pn) follows from an a priori L∞(0, T ;L2(Ω)) estimate, obtained as in [ALpr]. It is at this

level that we also find that B(v) ∈ L∞(0, T ;L1(Ω)) , which we have required in Definition 1.

The convergence of Galerkin approximations for (Pn) follows as in Theorem 1 and yields

existence of a weak solution vn .

Applying Theorem 1 to the constructed sequence vn , we get existence for (P ) . ⋄‡This hypothesis actually seems amount to nothing more than this inequality. Moreover, a natural candidate

for ϕ is bt , and this inequality, with (H3) and (H4) taken into account, means that it could be considered

as a part of the second member in (P ) . See Lemma 3 for a discussion of the condition (H13) in the scalar

case.

2.I.2. Proof of the continuity theorem 109

2 Proof of the continuity theorem

In order to prove Theorem 1, we start with the following lemma:

Lemma 2 Assume (P ) satisfy (H1), (H2), (H3), (H6), (H8) and Ψ(0, u0) ∈ L1(Ω) ; let

v is a weak solution of (P ) . Then

(i) supτ∈[0,T ]

∫

Ω

B(v)(τ, ·) + ‖v‖Lp(0,T ;W 1,p(Ω; IRN )) ≤M

(ii) there exists a function ω ∈ C(IR+; IR+) , ω(0) = 0 such that for all E ⊂ Q

∫∫

E

|b(v)| ≤ ω(|E|);

here M and ω(·) are determined solely by ‖Ψ(0, u0)‖L1(Ω) , c,K0, K1, K2 in (5) and the

dependence of Kε, Kε2 in (6) on ε .

Proof of Lemma 2: Take vχ[0,t)×Ω for the test function in (ii2) of Definition 1. Applying

Lemma 1, on account of (H3), (H6), (H8) we get for a.a. t ∈ (0, T )

∫

Ω

B(v)(t) −∫

Ω

Ψ(0, u0) + c

∫ t

0

∫

Ω

|Dv|p ≤ ε(∫ t

0

∫

Ω

|v|p +

∫ t

0

∫

∂Ω

|v|p)+

+

∫ t

0

Kε1(τ)

∫

Ω

B(v)(τ) dτ +

∫ t

0

∫

Ω

Kε2 +

∫ t

0

∫

∂Ω

Kε3

with ε > 0 and the corresponding Kε1 , K

ε2, K

ε3 . Hence, using the imbedding of W 1,p(Ω)

into Lp(∂Ω) and the Poincare inequality, we get for all ε sufficiently small

∫

Ω

B(v)(t) +c

2‖v‖Lp(0,T ;W 1,p(Ω;IRN )) ≤ ‖Ψ(u0)‖L1(Ω) +

+

∫ t

0

Kε1(τ)

∫

Ω

B(v)(τ) dτ + ‖Kε2‖L1(Q) + ‖Kε

3‖L1(Σ).

Thus (i) follows from the Gronwall inequality.

Consequently, (ii) follows readily from (7); it suffices to take

ω(r) = minδ>0

(Mδ + sup

E⊂Q,|E|≤r

∫∫

E

K0(·, 1/δ)). ⋄

Now let prove the compactness part of the continuity theorem.

Proof of (i) in Theorem 1: By Lemma 2, ‖vn‖Lp(0,T ;W 1,p(Ω; IRN )) are bounded;

moreover, un = bn(vn) are equiintegrable on Q .

Let prove the “compactness in x ” of un in L1(Q) (i.e. the property (10) below); its

compactness in (x, t) in L1(Q) will follow from Lemma 6 (see the Appendix and [K69a]).

Indeed, un = bn(vn) satisfies the evolution equation ∂/∂t un = Fn in D′(Q) , with Fn

bounded in L1(0, T ;W−1,1(Ω)) by virtue of (H4), (H7), (H8) and (i) of Lemma 2. More-

over, un is bounded in L1(Q) by (ii) of Lemma 2. So we only need to show that for all

compact set K ⊂ Ω , for all h ∈ [0, dist(K, ∂Ω)/2] one has

In = sup|∆x|≤h

∫ T

0

∫

K

|un(t, x+∆x)− un(t, x)∣∣∣ dxdt ≤ ωK(h) (10)

with some function ωK ∈ C(IR+; IR+) with ωK(0) = 0 , ωK independent of n .

Fix α > 0 . First, by Remark 3 and the Lusin and Egorov theorems there exists an

open set Qα ⊂ Q , |Qα| < α such that bn → b in C(Q \ Qα, C(IRN ; IRN )) = C((Q \

Qα) × IRN ; IRN) . Thus it follows by the Arzela-Ascoli theorem that there exists a function

ωα,M ∈ C((IR+)3; IR+) with ωα,M(0, 0, 0) = 0 such that for all (t, x), (t′, x′) ∈ Q \Qα , all

z, z′ ∈ [−M,M ]N one has |bn(t, x, z)− bn(t′, x′, z′)| ≤ ωα,M

(|t− t′|, |x− x′|, |z − z′|

).

Second, take M = M(α) = 1/α supn ‖vn‖L1(Q) ; by the Chebyshev inequality, for each

n ∈ IN there exists another set Qα,n ⊂ Q , |Qα,n| < α such that |vn| ≤ M(α) a.e. on

Q \Qα,n .

Now we can estimate In in (10) by integrating separately over the set Qα,n(∆x) =

(t, x) ∈ (0, T )×K : (t, x) ∈ Qα ∪Qα,n or (t, x+∆x) ∈ Qα ∪Qα,n , with |Qα,n(∆x)| <4α , and the complementary set Q′

α,n(∆x) = ((0, T )×K) \Qα,n(∆x) . By the concavity of

ωα,M(0, |∆x|, ·

), we get

In ≤ ω(4α) +

∫∫

Q′α,n(∆x)

|bn(t, x+∆x, vn(t, x+∆x))− bn(t, x, vn(t, x))| dxdt ≤

≤ ω(4α) + +

∫∫

Q

ωα,M(0, |∆x|, |vn(t, x+∆x)− vn(t, x)|

)≤ ω(4α) +

+ |Q|ωα,M(0, |∆x|, 1

|Q|

∫∫

Q

|vn(t, x+∆x)− vn(t, x)|)

≤

≤ ω(4α) + |Q|ωα,M(0, h, sup

n

‖Dvn‖L1(Q)

|Q|h).

Minimizing the right-hand side in α > 0 , we get a function ωK with the desired properties.

⋄

Finally, let prove the convergence part of the continuity theorem.

Proof of (ii) in Theorem 1: By compactness, choose a subsequence (which we still

denote by vn ) such that vn v in Lp(0, T ;V ) , vn → v in Lp(Σ; IRN) and a.e., and

bn(vn) → u in L1(Q; IRN ) and a.e. on Q .

It follows that u = b(v) , by the argument introduced in [BrSt73]. More precisely, by

Remark 3, the Egorov theorem and the Chebyshev inequality, for all α > 0 there exists an

open set Qα ⊂ Q such that |Qα| < α , bn → b in C(Q \ Qα;C(IRN)) , bn(vn) → u in

L∞(Q \Qα) , and v is bounded on Q \Qα . Therefore for all η, ζ ∈ L∞(Q \Qα) we have

bn(η) → b(η) in L∞(Q \Qα) ; in addition, b(v + λζ) → b(v) in L∞(Q \Qα) as λ→ 0 .

2.I.2. Proof of the continuity theorem 111

Thus, by the monotonicity of bn ,∫∫

Q\Qα

u(v − η) = lim

∫∫

Q\Qα

bn(vn)(vn − η) ≥

≥ lim

∫∫

Q\Qα

bn(η)(vn − η) =

∫∫

Q\Qα

b(η)(v − η).

Taking η = v + λζ with arbitrary ζ ∈ L∞(Q \ Qα) , letting λ decrease to 0 and then

increase to 0 , we conclude that u = b(v) in L1(Q \ Qα) . Letting α go to 0 , we get

u = b(v) a.e. on Q .

It follows that bn(vn)t → b(v)t in Lp′(0, T ;V ′) ; indeed, note that bn(vn)t are uni-

formly bounded in this latter space by (ii2) of Definition 1, Lemma 2 and hypotheses

(H4), (H7), (H8) .

The initial condition (ii1) is therefore satisfied at the limit; indeed, (9) implies that

u0,n → u0 in L1(Ω; IRN) . Note also that, by the Fatou lemma, B(v) ∈ L∞(0, T ;L1(Ω)) ,

which will permit to apply Lemma 1 to the function v .

Let prove that (ii2) holds as well.

Start by showing that one has gn(vn) → g(v) and fn(vn) → f(v) in Lp′(Σ; IRN) and

Lp′(Q; IRN) , respectively. The former follows readily from (H8) and the Lebesgue dominated

convergence theorem. We need the structure condition (H11) in order to prove the latter.

Let us fix ǫ > 0 and show that, for all n sufficiently large, one has∫∫

Q

|fn(vn)− f(v)|p′ < ǫ. (11)

First, it follows from (H4) and Lemma 2 that |fn(vn)|p′ are equiintegrable on Q .

Indeed, take α > 0 . For all E ⊂ Q , |E| < α , we have for ε > 0 and the corresponding

Kε2

∫∫

E

|fn(vn)|p′ ≤ ε

(∫ T

0

∫

Ω

Kε2(t)Bn(vn)(t)dt+

∫∫

Q

|vn|p)+

∫∫

E

K2 ≤

≤ ε supn

(‖K1‖L1(0,T ) ‖Bn(vn)‖L∞(0,T ;L1(Ω)) + ‖vn‖pLp(0,T ;V )

)+

∫∫

E

Kε2 ,

which is independent of n and can be made as small as desired by a choice of ε and α

small enough. Fix α > 0 such that

∫∫

E

|fn(vn)− f(v)|p′ < ǫ/3 whenever |E| < α .

Further, by the Chebyshev inequality, there exists M > 0 such that for all n ∈ IN one

can choose an open set Qα,n ⊂ Q with |Qα,n| < α so that |vn| ≤ M , |v| ≤ M on

Q \Qα,n . Fix M ; for all (t, x) ∈ Q define the set Kt,xM ⊂ IRN as the image of [−M,M ]N

by b(t, x, ·) . Let TM (t, x, ·) be the projection of IRN on this set, i.e.,

TM(t, x, ·) : z ∈ IRN 7→ TM(t, x, z) = z ∈ Kt,x

M ,

where dist(z, z) = minζ∈Kt,xM

dist(z, ζ).(12)

The projection is well defined since, clearly, Kt,xM is compact and convex. Note that TM :

Q× IRN 7→ IRN is Caratheodory. Indeed, it is a contraction in IRN for all (t, x) ∈ Q ; for

all z ∈ IRN it is measurable in (t, x) , because b is measurable in (t, x) and TM depends

continuously on b with respect to the norm ‖ · ‖C([−M,M ]N) .

Now for all n ∈ N , for a.a. (t, x) ∈ Q , TM bn = TM(t, x, bn(t, x, ·)) is well defined.

By (H11) , one has for a.a. (t, x) ∈ Q

|fn(vn)− f(v)|p′ ≤ const( ∣∣∣fn(vn)− f(vn)

∣∣∣p′

+

+∣∣∣(f b)(vn)− (f TM b)(vn)

∣∣∣p′

+∣∣∣(f TM b)(vn)− (f TM bn)(vn)

∣∣∣p′

+

+∣∣∣(f TM )(bn(vn))− (f TM )(b(v))

∣∣∣p′

+∣∣∣(f TM b)(v)− (f b)(v)

∣∣∣p′ )

.

(13)

By Remark 3, we can apply the Lusin and Egorov theorems to fn, f and f TM bn, f TM b . Indeed, by (9) fn → f and f TM bn → f TM b in C([−M,M ]; IRN ) for a.a.

(t, x) ∈ Q . Besides, one also have bn(vn) → b(v) a.e. on Q . It follows that there exists

an open set Qα ⊂ Q with |Qα| such that one has

fn → f and f TM bn → f TM b in C((Q \Qα)× [−M,M ]N ; IRN );

bn(vn) → b(v) in C(Q \Qα; IRN );

f TM is uniformly continuous on (Q \Qα)× [−M,M ]N .

(14)

In addition, for a.a. (t, x) ∈ Q , TM b ≡ b on [−M,M ]N .

For each n , we can integrate in (11) separately over Qα ∪Qα,n and Q \ (Qα ∪Qα,n) .

The second integral vanishes as n→ ∞ , due to (13) and (14), and the first one is estimated

by 2ǫ/3 . Hence (11) holds for n small enough.

It remains to justify the passage to the limit in the elliptic term. This can be made through

a usual Minty-Browder argument summarized in Lemma 7 in Appendix. Let introduce the

operators An : η ∈ Lp(0, T ;V ) 7→ Lp′(0, T ;V ′) by defining the duality product of Anη with

ϕ ∈ Lp(0, T ;V ) :

∫ T

0

< Anη, ϕ >=

∫∫

Q

an(vn, Dη) : Dϕ.

By (H7) and Lemma 2, this last integral makes sense. Likewise, the operator A : η ∈Lp(0, T ;V ) 7→ Aη ∈ Lp′(0, T ;V ′) is defined by assigning

∫ T

0

< Aη, ϕ >=

∫∫

Q

a(v,Dη) :

Dϕ. . Since vn is a weak solution of (Pn) , it follows from the analysis above that Anvn∗ χ

in Lp′(0, T ;V ′) , where

∫ T

0

< χ, ϕ >=

∫ T

0

< −b(v)t, ϕ > +

∫∫

Q

f(v) · ϕ+

∫∫

Σ

g(v) · ϕ

for ϕ ∈ Lp(0, T ;V ) .

2.I.3. Comments and further results 113

Let verify the other assumptions of Lemma 7. We have vn v in Lp(0, T ;V ) ; all

An are monotone; besides, A is hemicontinuous by (H7) and the Lebesgue dominated

convergence theorem. Further, under the assumption (H12) , Anη → Aη in Lp′(0, T ;V ′)

for all fixed η ∈ Lp(0, T ;V ) ; the arguments are the same as used for the proof of convergence

of fn(vn) above.

Furthermore, let prove that one also has

lim inf

∫ T

0

< Anvn, vn >≤∫ T

0

< χ, v > . (15)

First, under the structure condition (H13) we can show that

∫∫Φn,t(vn) →

∫∫Φt(v) as

n→ ∞ (the last integral makes sense, by Lemma 1). Indeed, by (H13) Φt(v) = φ(b(v))+

v · ϕ(b(v)) and ϕ(b(v)) ∈ Lp′(Q) . Since vn v in Lp(Q) , it suffices to show that∫∫

Q

|Φn,t(vn)− φ(b(v))− vn · ϕ(b(v))| vanishes as n→ ∞ . This can be done by the same

arguments as used for the proof of convergence of fn(vn) . With TM(t, x, ·) defined by (12),

we have to replace the key estimate (13) by the lengthy, but trivial estimate

∣∣∣Φn,t(vn)− φ(b(v))− vn · ϕ(b(v))∣∣∣ ≤

∣∣∣Φn,t(vn)− Φt(vn)∣∣∣ +

∣∣∣φ(b(vn))− φ(b(v))∣∣∣ +

+∣∣∣vn∣∣∣∣∣∣ϕ(b(vn))− ϕ(b(v))

∣∣∣ ≤∣∣∣Φn,t(vn)− Φt(vn)

∣∣∣ +∣∣∣(φ b)(vn)− (φ TM b)(vn)

∣∣∣ +

+∣∣∣(φ TM b)(vn)− (φ TM bn)(vn)

∣∣∣ +∣∣∣(φ TM)(bn(vn))− (φ TM)(b(v))

∣∣∣ +

+∣∣∣(φ TM b)(v)− (φ b)(v)

∣∣∣ + ǫ∣∣∣vn∣∣∣p

+ C(ǫ)

∣∣∣(ϕ b)(vn)− (ϕ TM b)(vn)∣∣∣p′

+

+∣∣∣(ϕ TM b)(vn)− (ϕ TM bn)(vn)

∣∣∣p′

+∣∣∣(ϕ TM)(bn(vn))− (ϕ TM)(b(v))

∣∣∣p′

+

+∣∣∣(ϕ TM b)(v)− (ϕ b)(v)

∣∣∣p′,

where we have used the inequality |a · b| ≤ ǫ|a|p + C(ǫ)|b|p′ for a, b ∈ IRN . The required

convergence will follow from the convergences of bn,Φn,t to b,Φt , respectively, given by (9),

and the a.e. convergence of bn(vn) to b(v) . Secondly, without loss of generality we can

apply Lemma 1 with t = T to v and all functions vn , n ∈ IN . Using the convergence of

Ψn(0, u0,n) in (9), we have

lim supn→∞

∫ T

0

< −bn(vn)t, vn >= lim supn→∞

(−∫

Ω

Ψn(bn(vn))(T ) +

∫

Ω

Ψn(0, u0,n))−

−∫∫

Q

Φn,t(vn) ≤ −∫

Ω

Ψ(b(v))(T ) +

∫

Ω

Ψ(u0)−∫∫

Q

Φt(v) =

∫ T

0

< −b(v)t, v >;

the inequality here is due to the Fatou Lemma. Together with the strong convergence of

gn(vn) and fn(vn) , this yields (15). By Lemma 7, χ = Av in Lp′(0, T ;V ′) , which implies

(ii2) . ⋄


3 Comments and further results

3.1. On the inhomogeneous Dirichlet boundary conditions

One can deduce from Corollary 1 some existence results for non-zero Dirichlet boundary con-

ditions h ∈ Lp(0, T ;W 1,p(Ω; IRN)) . Indeed, Definition 1 permits to perform the translation

Th : v 7→ v − h ; writing down the restrictions induced by (H2)−(H13) and Th , we obtain

hypotheses that guarantee the existence of weak solutions. Within certain classes of h , the

hypotheses of Section 1 remain invariant. Let give two examples. For simplicity, assume that

K1, Kε1 ∈ L∞(0, T ) for the first case.

Corollary 2 Let (H1), (H4)−(H9), (H11)−(H13) hold, with K1 = const in (H4), (H7) ,

(H13) and Kε1 = constε in (H6) for each ε > 0 . Assume that for all z ∈ IRN , all

t ∈ [0, T ] and a.a. x ∈ Ω

|b(t, x, z)| ≤ K0(t, x)(1 + |z|p/κ

), where κ ∈ [1, p), κ′ = κ/(κ− 1)

and K0 ∈ Lν(Q) with K0(0, ·) ∈ Lν(Ω) for some ν ∈ [κ′,+∞];(H2’)

assume that for all ε > 0 there exists Kε2 ∈ L1(Q) such that for all z ∈ IRN and a.a.

(t, x) ∈ Q

|Φt(t, x, z)| ≤ ε|z|p +Kε2(t, x). (H3’)

Let 1/σ + 1/ν + 1/κ = 1 . Then for all Dirichlet data h ∈ Lp(0, T ;W 1,p(IRN )) such that

h ∈ W 1,σ(0, T ;Lσ(Ω; IRN)),

h(·, x) is absolutely continuous on [0, T ] for a.a. x ∈ Ω

with h(0, ·) ∈ Lp(Ω; IRN) and h(0, ·) · u0(·) ∈ L1(Ω),

(H10’)

there exists a weak solution to (P ) .

Proof of Corollary 2: It is equivalent to consider instead of (P ) the problem (P )

with the zero Dirichlet data and b(t, x, z) = b(t, x, z + h(t, x)) , a(t, x, z, ξ) = a(t, x, z +

h(t, x, ), ξ +Dh(t, x)) , f(t, x, z) = f(t, x, z + h(t, x)) and g(t, x, z) = g(t, x, z + h(t, x)) .

Note that (H1) holds with Φ(t, x, z) = Φ(t, x, z + h(t, x)) − Φ(t, x, h(t, x)) , so that

Ψ(t, x, z∗) = Ψ(t, x, z∗)− h(t, x) · z∗ + Φ(t, x, h(t, x)) and

B(t, x, z) = B(t, x, z + h(t, x))− h(t, x) · b(t, x, z + h(t, x)) + Φ(t, x, h(t, x)). (16)

Clearly, the initial condition is unchanged: u0 = u0 on Ω .

Since h(0, ·) ∈ Lσ(Q; IRN) by the Fatou lemma, and because

|Φ(0, x, h(0, x))| ≤ K0(0, x) |h(0, x)| (1 + |h(0, x)|p/κ) ≤≤ const

((K0(0, x))

ν + |h(0, x)|σ + 1 + |h(0, x)|p),


it follows from (H2′) and (H10′) that Ψ(u0) ∈ L1(Ω) . Hence (H9) holds. The hy-

pothesis (H2) is satisfied because K0 |h|p/κ ∈ L1(Q) . Further, (H5), (H11), (H12) are

obvious. Since h,Dh ∈ Lp(Q) , the invariance of (H8) follows by the Holder inequality;

moreover, checking (H4), (H6), (H7) amounts to showing that

B(t, x, z + h(t, x)) ≤ B(t, x, z) + C|z|p +K2(t, x) (17)

with some function K2(t, x) ∈ L1(Q) . Since Φ is non-negative, one has by (H2′) and

(16)

B(t, x, z + h(t, x)) ≤ B(t, x, z) + h(t, x) · b(t, x, z + h(t, x)) ≤≤ B(t, x, z) + |h(t, x)| |K0(t, x)| (1 + |z + h(t, x)|p/κ),

(18)

whence (17) follows.

It remains to show that (H3) and (H13) hold for (P ) as well. One has

Φt(t, x, z) = ∂/∂t h(t, x) ·(b(t, x, z + h(t, x))− b(t, x, h(t, x)

)+

+Φt(t, x, z + h(t, x))− Φt(t, x, h(t, x)).(19)

Hence Φt(t, x, z) = (φ)(t, x, b(t, x, z)) + z · (ϕ)(t, x, b(t, x, z)) with (ϕ) ≡ ϕ . As in the

estimate (18), using in addition the inequality c|a · b| ≤ C(ǫ)cν +C(ǫ)|a|σ+ ǫ|b|κ valid for all

c ∈ IR+, a, b ∈ IRN , one sees that (H3′) and the inequality in (H13) are invariant under

the translation by h , whenever (H2′), (H10′) hold.

Therefore (P ) is in the scope of Corollary 1. ⋄

Remark 4 Clearly, under the assumptions (H1), (H2′), (H3′), (H4)−(H9), (H11)−(H13)

on (P ) , for the class of Dirichlet data h verifying (H10′) there is also a continuous depen-

dence on h of weak solutions of (P ) in the sense of Theorem 1.

In case without dependence of coefficients on (t, x) , the assumptions on h of Corollary 2

meet exactly the assumptions in the remark in [AL83] that follows Lemma 1.5. Nevertheless,

if we could modify the condition (H3) by excluding s 6= t (cf. Remark 6 in the Appendix),

there would be no need to strengthen (H3) to (H3′) . The problem here is that, in what

concerns s 6= t , (H3) is utterly non-invariant under translations Th with natural assump-

tions on h . For instance, with the hypothesis (H3) we cannot attain the assumptions on h

made in the statement of Lemma 1.5 in [AL83] unless requiring (H2′′) below, which imposes

a growth restriction in z †. We state the corresponding result in case where b is independent

of t , so that (H3) does not seem pecular.

†This hypothesis is verified up to exponential growthes of b(x, z) in z

Corollary 3 Let b be independent of t , and (H1), (H3)−(H9), (H11), (H12) hold. As-

sume

for all r ∈ IR+ there exist Λ(r) <∞ and K0(·, r) ∈ L1(Ω)

such that for all λ, z ∈ IRN , and a.a. x ∈ Ω one has

|b(x, z + λ)| ≤ Λ(|λ|)|b(x, z)|+ K0(x, |λ|) and |b(x, 0)| ≤ K0(x, 0).

(H2”)

Then there exists a weak solution of (P ) for all Dirichlet data h ∈ Lp(0, T ;W 1,p(IRN)) such

that

h ∈ L∞(Q; IRN), ∂/∂t h ∈ L1(0, T ;L∞(Ω; IRN ))

and h(·, x) is absolutely continuous on [0, T ] for a.a. x ∈ Ω.(H10”)

Proof of Corollary 3: We proceed as in the proof above. First note that it follows

from (H2′′) that |b(x, z)| ≤ K0(x, |z|) = Λ(|z|)K0(x, 0)+K0(x, |z|) . Let M = ‖h‖L∞(Q) .

Clearly, |b(t, x, z)| ≤ K0(x, |z| +M) , so that (H2), (H9) hold for (P ) . Besides, one has

to show that

|Φt(t, x, z)| ≤ εK1(t)B(s, x, z) +Kε2(t, x) (20)

for all z ∈ IRN , a.a. t, s ∈ (0, T ) and a.a. x ∈ Ω , where K1 ∈ L1(0, T ) and Kε2(t, x) ∈

L1(Q) , and that

B(x, z + h(t, x)) ≤ B(t, x, z) +K2(t, x) (21)

for all z ∈ IRN and a.a. (t, x) ∈ Q , where K2 ∈ L∞(0, T ;L1(Ω)) .

First, by (H2′′) and (7) with δ = ε/Λ(2M) one has

|b(x, z + h(t, x))| ≤ Λ(2M)|b(x, z + h(s, x))|+ K0(x, 2M) ≤ Λ(2M)|b(s, x, z)|++K0(x, 2M) ≤ εB(s, x, z) +K0(x, ε/Λ(2M) +M) + K0(x, 2M).

Hence it follows from (19), with Φt ≡ 0 , that (20) holds with K1(t) = ‖∂/∂t h(t, ·)‖L∞(Ω)

and Kε2(t, x) = |∂/∂t h(t, x)|

(K0(x,M) +K0(x, ε/Λ(2M) +M) + K0(x, 2M)

).

Further, (21) holds, because by (16) and (7) with δ = 1 one has

B(x, z + h(t, x)| ≤ B(t, x, z) + |h(t, x)| |b(t, x, z)| ≤ (1 +M)B(t, x, z) + K0(x,M + 1). ⋄

3.2. On the structure conditions

Let us turn now to the relevancy of the hypotheses (H11)−(H13) . First note one direct

generalization of Theorem 1 and Corollaries 1,2,3.

Remark 5 An additional term of the form f(·, v) can be considered in the right-hand side

of (P ) , provided f : Q× IRN 7→ IRN is Caratheodory and satisfy

(f(t, x, z)− f(t, x, z)) · (z − z) ≤ 0,

|f(t, x, z)|p′ ≤ C |z|p +K1(t)B(t, x, z) +K2(t, x)(H14)

for all z, z ∈ IRN , for a.a. (t, x) ∈ Q , with some K2 ∈ L1(Q) and C = const ≥ 0 . For

(P ) modified in such a way, (H) and (9) modified correspondingly, and (H14) admitted,

Theorem 1 and Corollary 1 still hold. Indeed, in this case we can include the terms −fn(t, x, ·)into the operators An , and An remain monotone. Therefore we do not need a structure

condition of the kind (H11) on f .

Besides this remark, it is easy to see that in case the dependence on z of f and a is

weak, (H11) and (H12) can be superfluous while proving the existence of a weak solution.

Indeed, for p = 2 consider the model problem

b(v)t = ∆v + divF (v) on Q = (0, T )× Ω (MP)

with F : IRN 7→ IRN which is globally Lipschitz continuous, and with the homogeneous

Dirichlet boundary condition (so that V = W 1,p0 (Ω; IRN ) ). The operator A : η ∈ L2(0, T ;V )

7→ L2(0, T ;V ′) defined by∫ T

0

< Aη, ϕ >=

∫∫

Q

Dη : Dϕ+ F (η) ·Dϕ

for all ϕ ∈ L2(0, T ;V ) is monotone provided the diameter of Ω is small, due to the Poincare

inequality. Therefore we can use Lemma 7 with An ≡ A while passing to the limit in a

sequence of approximating problems (MPn) with bn bilipschitz, and with solutions vn such

that vn v in L2(0, T ;V ) . This yields that Avn Av in L2(0, T ;V ′) , but also that

vn → v strongly in L2(0, T ;V ) ; so one can pass to the limit in each term in (MP ) and

infer the existence.

For scalar problems of type (MP ) , an existence result for arbitrary locally Lipschitz

continuous F has been shown in [BW99]. The methods used in this paper are those of the

nonlinear semigroup theory; unfortunately, this gives no information on the type of convergence

of approximating solutions to the weak solution of (MP ) .

Practically, (H11) , (H12) often hold in problems motivated by physics. For instance, in

the Richards equation (cf. [Bear]) v and u represent the liquid pressure and the saturation of

the medium, respectively; the nonlinearity in the right-hand side of the corresponding equation

is the permeability of the medium which is a function of u . On the other hand, the condition

(H13) is proper to the function b(·) and seems to impose restrictions on its dependence on

t . While the author was not able to prove that the representation

Φt(t, x, z) = φ(t, x, b(t, x, z)) + z · ϕ(t, x, b(t, x, z))for all z ∈ IRN and a.a. (t, x) ∈ Q

(22)


with φ, ϕ Caratheodory generically holds, the following result fixes the situation in the scalar

case.

Lemma 3 Let N = 1 . Assume that bt is Caratheodory. Then (22) holds, and one can take

ϕ such that for all z ∈ IRN and a.a. (t, x) ∈ Q one has ϕ(t, x, b(t, x, z)) = bt(t, x, z) .

Proof: The dependence on x is immaterial here, and we neglect it in the notation.

Let first prove that, for a.a. t0 ∈ (0, T ) , if Φ(t0, ·) is affine in an interval [z, z] (i.e.,

b(t0, ·) degenerates) it follows that its derivative with respect to t is also affine on [z, z] .

Indeed, assume that this property fails on some subset B of (0, T ) with |B| > 0 . For each

t0 ∈ B there exist two rational points z, z ∈ [z, z] such that the function

∆(t0, z, z; ·) = (z − z)(Φ(t0, ·)− Φ(t0, z))− (· − z)(Φ(t0, z)− Φ(t0, z))

is identically zero on [z, z] , while the continuous function

∇(t0, z, z; ·) = (z − z)(Φt(t0, ·)− Φt(t0, z))− (· − z)(Φt(t0, z)− Φt(t0, z))

is non-zero at some rational point z0 ∈ (z, z) . Still ∇(t0, z, z; z0) = ∆t(t0, z, z; z0) . Hence

B ⊂ ⋃Bz,z,z0 , where Bz,z,z0 = t0 ∈ (0, T ) | ∆(t0, z, z; z0) = 0, but ∆t(t0, z, z; z0) 6= 0 ,

and the union is taken over all z, z, z0 ∈ IQ . Since ∆(·, z, z; z0) is absolutely continuous on

[0, T ] , |Bz,z,z0| = 0 for all z, z, z0 ∈ IQ , which is a contradiction.

Note that for t0 /∈ B , one has bt(t0, ·) = (Φt(t0, z) − Φt(t0, z))/(z − z) = const on

[z, z] . Hence we have shown that for a.a. (t, x) ∈ Q , all z, z ∈ IRN one has

b(t, z) = b(t, z) ⇒

bt(t, z) = bt(t, z) and

Φt(t, z)− Φt(t, z) = (z − z)bt(t, z)

for all z between z and z.

(23)

This is equivalent to the statement of the lemma. Indeed, (23) means that bt(t, ·) is in

fact a function of b(t, ·) : bt(t, z) = ϕ(t, b(t, z)) . The function ϕ(t, ·) is continuous. Indeed,

let βn, β ∈ R(b(t, ·)) and βn → β as n → ∞ . Due to the monotony of b(t, ·) , thereexist zn ∈ b(t, ·)−1(βn) , z ∈ b(t, ·)−1(β) such that zn → z . Hence ϕ(t, βn) = bt(t, zn) →bt(t, z) = ϕ(t, β) as n→ ∞ . Furthermore, by (23), Φt(t, ·)− z ϕ(t, b(t, ·)) stays constant

as b(t, ·) degenerates. By the same argument, it can be written under the form φ(t, b(t, ·)) ,with φ(t, ·) continuous. ⋄

3.3. On pure Neumann boundary conditions for one or more components

The restriction |Γ0| > 0 , imposed in Sections 1,2, can be relaxed in two ways. The first

one consists in imposing hypotheses that provide a priori estimates on ‖Dv‖Lp(Q) and

‖B(v)‖L∞(0,T ;L1(Q)) , and another hypothesis that permits to control |v| by |B(v)| . This

requires some modification of our arguments, and we do not pursue this line here. On the

2.I.4. Appendix 119

other hand, some cases can be included by requiring directly the coercivity of the elliptic part

in the space V for all t ∈ (0, T ) ; more precisely, it is sufficient (in case h ≡ 0 ) that, instead

of (H6) and the second part of (H8) , the following assumption be fulfilled:∫

Ω

a(t, ·, w(·), Dw(·)) : Dw(·)− f(t, ·, w(·)) · w(·) + Φt(t, ·, w(·))

−

−N∑

i=1

∫

Γ1,i

gi(t, ·, w(·))wi(·) ≥ c‖w‖V −K1(t)B(t, ·, w(·))−K1(t)

for all w ∈ V and a.a. t ∈ (0, T ) , where K1 ∈ L1(0, T ) and c > 0 . In this case, Lemma 2

is immediate, and other arguments run without any modification.

3.4. On the convergence of approximate methods

Note that Theorem 1 can be used to prove the convergence while solving (P ) by various

approximate methods. One can almost directly apply the theorem to Galerkin approximations.

Indeed, the equation for Galerkin approximations is (P ) itself with an error term that tends

to zero weakly in Lp′(0, T ;V ′) , which still permits to apply the Minty-Browder argument

for convergence. For finite volumes methods, an example is given in [BAGuW] and Chapter

2.II, where the convergence for a class of schemes is proved for a simple model elliptic-

parabolic system. The steps of the proof of Theorem 1 are the same, provided the system that

determines the discret solution is rewritten under the form of equation in D′ similar to (P ) .

Approximations by time implicit discretization (at least, with uniform step) can be treated in

a similar way; in this case writing the system that determines the solution under the form of

equation in D′ is trivial.

Appendix

Here we present the appropriate versions of the three essential arguments involved in the proof

of Theorem 1.

A1. The chain rule argument

Let use the shortened notation w(t) for w(t, x) , b(w)(t) for b(t, x, w(t, x)) , Φt(w)(t) for

Φt(t, x, w(t, x)) , Φt(τ, w(t)) for Φt(τ, x, w(t, x)) and so on. Let start with two auxiliary

results.

Lemma 4 For k > 0 , let Tk : z ∈ IRN 7→ min|z|, k z/|z| ∈ IRN . Assume that b satisfy

(H1) and Φ is absolutely continuous on [0, T ] for all z ∈ IRN and a.a. x ∈ Ω . Then one

has (omitting the dependence in x )

B(t, Tk(z)) ≤ B(s, z) + (b(t, z)− b(s, z)) · Tk(z)−∫ t

s

Φt(τ, Tk(z)) dτ

for all z, z ∈ IRN , t, s ∈ [0, T ] , x ∈ Ω .

Proof: By the convexity of Φ in z , one has

B(t, Tk(z))− B(s, z) = B(s, Tk(z))−B(s, z) +B(t, Tk(z))− B(s, Tk(z)) ≤≤ (b(s, Tk(z))− b(s, z)) · Tk(z) + b(t, Tk(z)) · Tk(z)− b(s, z) · Tk(z)−

−Φ(t, Tk(z)) + Φ(s, Tk(z)) = (b(t, Tk(z))− b(s, z)) · Tk(z)−∫ t

s

Φt(τ, Tk(z)) dτ.

Since b is monotone and z−Tk(z) = (|z|/k−1)+Tk(z) , it follows that b(t, Tk(z)) ·Tk(z) ≤b(t, z) · Tk(z) . ⋄

Lemma 5 Let φ : Q × IRN 7→ IR be Caratheodory. Assume that for all z ∈ IRN , for a.a.

(t, x) ∈ Q one has

|φ(t, x, z)| ≤ F (t, x, |z|) with F (·, x, r) ∈ L1(0, T ) for all r > 0. (24)

For h 6= 0 , set φh(t, x, z) = 1/h

∫ t

t−h

φ(τ, x, z) dτ , where φ(·, z) is extended by 0 outside

Q . Assume that v : Q 7→ IRN is measurable and for all h one has

|φh(v)(·)| ≤ F (·) for all h, where F ∈ L1(Q). (25)

Then φh are Caratheodory and φh(v) → φ(v) in L1(Q) .

Proof: By (24), for all h > 0 , for all bounded set K in IRN , |φh(·, z)| are dominated

by a fixed function in L1(Q) for all z ∈ K . Therefore φh are Caratheodory.

By Remark 3 it follows that φ, φh are strongly measurable from Q to C(IRN) . Moreover,

φh → φ a.e. on Q in C(IRN) . Therefore for all sequence hk → 0 one can apply the Lusin

and Egorov theorems to φhk . Thus for all α > 0 there exists an open subset Qα of Q

with |Qα| < α such that φhk → φ in C(Q \Qα;C(IRN)) = C(Q \Qα × IRN) .

By (25), φhk(v) are equiintegrable on Q , so that it is sufficient to prove the lemma in

case v(·) is bounded on Q by some constant M and φhk , φ are equicontinuous in (t, x, z)

on Q × [−M,M ]N . Let approximate v a.e. on Q by a sequence of functions vm each

taking a finite number of values in [−M,M ]N . By the diagonal process, one can extract a

subsequence hkn such that φhkn (vm) → φ(vm) a.e. on Q for all m ∈ IN . Take (t, x) in

the set where this convergence takes place. For each m fixed, one has

|φhkn (v)− φ(v)| ≤ |φhkn (v)− φhkn (vm)|+ |φhkn (vm)− φ(vm)|+ |φ(vm)− φ(v)|,

which can be made as small as desired by a choice of m and n large enough. Since the

sequence hk → 0 is arbitrary, the assertion of the lemma follows by the Lebesgue theorem.

⋄

Proof of Lemma 1: Take v0 measurable such that b(v0) = u0 , and extend v(t) by v0

to t < 0 . One has b(v)t = −χ ≡ 0 for t < 0 .

2.I.4. Appendix 121

First, take t ∈ (0, T ) , h > t − T and apply Lemma 4 to s = t − h , z = v(t) ,

z = v(t− h) . Denote Tk(v(·)) by vk(·) ; note that ‖vk(t)‖V ≤ ‖v(t)‖V . After integration

in x , one obtains∫B(vk)(t) ≤

∫B(v)(t− h)+

+

∫(b(v)(t)− b(v)(t− h)) · vk(t)−

∫ t

t−h

∫Φt(τ, vk(t)) dτ.

(26)

By (ii1) of Definition 1 and (H3) we deduce

∫ T

0

(∫B(v)(t)−

∫B(v)(t− h)

)dt ≤ const |h|

∫ T

0

1h

∫ t

t−h

‖χ(τ)‖V ′ dτ ‖v(t)‖V +

+1

h

∫ t

t−h

K1(τ) dτ

∫B(v)(t) +

∫|v(t)|p +

1

h

∫ t

t−h

∫Kε

2(τ) dτdt ≤

≤ const |h|‖χ‖Lp′(0,T ;V ′) ‖v‖Lp(0,T ;V ) + ‖K1‖L1(0,T )×

×‖B(v)‖L∞(0,T ;L1(Ω)) + ‖v‖pLp(Q) + ‖Kε2‖L1(Q)

≤ const |h|.

Therefore

∫B(v) ∈ W 1,1(0, T ) . In particular, there exists l = lim

h↓0

∫B(v)(h) . We have

to show that l =

∫B(v0) . Take t = 0 , z = v0 , s = h , z = v(h) in Lemma 4. By

Definition 1 (ii1) and (H3) one has

∫B(v)(h) ≥

∫B(Tk(v0))−

∫ h

0

(u0 − b(v)(h)) · Tk(v0)−∫ h

0

∫Φt(τ, Tk(v0)) dτ ≥

≥∫B(Tk(v0)) −

∫ h

0

‖χ(τ)‖V ′ dτ ‖Tk(v0)‖V −∫ h

0

K1(τ) dτ×

×‖B(0, Tk(v0))‖L1(Ω) − ‖Tk(v0)‖pLp(Q) −∫ h

0

‖Kε2(τ)‖L1(Ω) dτ.

As h ↓ 0 and k → +∞ , one gets l ≥∫B(v0) by the Fatou lemma. The inverse inequality

can be obtained in a similar way from (26) with t = h .

To prove (8), it remains to show that

d

dt

∫

Ω

B(v) = − < χ(t), v(t) > −∫

Ω

Φt(v)(t).

By (26), it suffices to show that there exists a sequence hn ↓ 0 such that

1

hn

∫(b(v)(t)− b(v)(t∓ hn)) · v(t) =

1

hn< b(v)(t)− b(v)(t∓ hn), v(t) >→ ± < χ(t), v(t) >

in L1(Q) , and

1

hn

∫ t

t∓hn

∫Φt(τ, v(t)) dτ → ±

∫Φt(v)(t)


in L1(0, T ) . Since (b(v)(·) − b(v)(· ∓ hn))/hn → ±χ(·) in Lp′(0, T ;V ′) , the former

convergence is clear. Besides, the latter one can be proved by applying Lemma 5 to φ = Φt .

Indeed, (25) follows from (H3) ; (24) follows from (H3) and the continuity of B in (t, z) .

This ends the proof. ⋄

Remark 6 Formally, the formula (8) would make sense under the weaker version of (H3) :

|Φt(t, x, z)| ≤ ε(K1(t)B(t, x, z) + |z|p

)+Kε

2(t, x). (H3?)

for a.a. (t, x) ∈ Q and all z ∈ IRN . On the other hand, except in the proof above, we

actually need this weaker form whenever (H3) is used. At the present stage, it is not clear to

the author whether the analogue of Lemma 1 does hold when (H3) is replaced by (H3?) .

A2. The compactness argument

In his paper [K69a] Kruzhkov has proved that, for a bounded weak solution of a general

evolution equation, the L1 -modulus of continuity in t can be estimated via the L1 -modulus

of continuity in x and an a priori bound on the right-hand side in some Sobolev space

L1(0, T ;W−m,1

). Here we present this lemma, with somewhat less restrictive assumptions,

as a kind of compactness result.

Lemma 6 Let Ω be an open subset of IRN , E be a set of indexes, and uε ∈ L1((0, T )×Ω ,

ε ∈ E , satisfy

∂

∂tuε(t, x) =

∑

|α|≤m

(−1)αDαAεα(t, x) in D′((0, T )× Ω).

Assume that for all compact set K ⊂ Ω the following estimates hold uniformly in ε :

(i) ‖uε‖L1((0,T )×K) ≤ CK ;

(ii)∑

|α|≤m ‖Aεα‖L1((0,T )×K) ≤ CK ;

(iii) for all h > 0 small enough sup|∆x|≤h

∫ T

0

∫

K

|uε(t, x+∆x)− uε(t, x)|dxdt ≤ ωK(h) ,

with limh→0 ωK(h) = 0 .

Then the family uεε∈E is relatively compact for the L1loc((0, T )× Ω) -topology.

Proof of Lemma 6: We only need to find a function ωK on IR+ such that limτ→0

ωK(τ) = 0

and I :=

∫ T−τ

0

∫

K

|uε(t + ∆t, x) − uε(t, x)|dxdt ≤ ωK(τ) for all ∆t ∈ [0, τ ] and ε ∈ E .

Indeed, the result will follow from (i),(iii) and the Kolmogorov theorem.

2.I.4. Appendix 123

Let revise the proof from [K69a]. Fix ε ∈ E , ∆t ∈ [0, τ ] . For t ∈ [0, T − τ ] set

wt(·) := uε(t+∆t, ·)− uε(t, ·) . For all ϕ ∈ D(K) we have∫

K

wt(x)ϕ(x)dx =

∫ t+∆t

t

∫

K

∑|α|≤m

Aεα(θ, x)D

αϕ(x)dxdθ. (27)

For each t ∈ [0, T−τ ] take for the test function ϕ(·) a regularisation of signwt(·) . More

precisely, let δ := dist(K, ∂Ω) and h ∈ (0, δ/2) . Set K−h = x ∈ K : dist(x, ∂K) ≥ hand denote by χK−h

(·) its characteristic function. Choose ρ ∈ C∞0 ([−1, 1]N) , ρ ≥ 0 such

that

∫ρ(σ)dσ = 1 , and take in (27) ϕt(x) = h−N

∫ρ(x− y

h

)signwt(y)χK−h

(y)dy . It

is clear that for |α| ≤ m , ‖Dαϕt‖L1([0,T ]×K) ≤ const · h−m uniformly in t and ε .

Note that I = I1+I2 , where I1 =

∫ T−τ

0

∫

K

wt(x)ϕt(x)dxdt and I2 =

∫ T−τ

0

∫

K

(|wt(x)|−

wt(x)ϕt(x))dxdt . Using (27) and (ii), we get |I1| ≤ const ·τh−m

∑|α|≤m ‖Aε

α‖L1([0,T ]×K) ≤

CKτ/h−m . Besides, by (i),(iii) and the Kolmogorov theorem the family

∫ T

0

|uε(t, ·)|dtε∈E

is relatively compact in L1(K) . Therefore these functions are equiintegrable on K , so that∫ T

0

∫

K\K−2h

|uε(t, x)|dxdt ≤ ωK(h) with limh→0 ωK(h) = 0 uniformly in ε . Hence

|I2| ≤ 2

∫ T−τ

0

∫

K\K−2h

|wt(x)|dxdt+

+

∫ T−τ

0

∫

K−2h

∣∣∣|wt(x)| − wt(x)

∫h−Nρ

(x− y

h

)signwt(y)

∣∣∣dydxdt ≤ 4ωK(h) +

+

∫ T−τ

0

∫

K−2h

∫h−Nρ

(x− y

h

) ∣∣∣|wt(x)| − wt(x) signwt(y)∣∣∣dydxdt.

Since for all a, b ∈ IR we have ||a| − a sign b| ≤ 2|a− b| , it follows that

|I2| ≤ 4ωK(h) + 2

∫ T−τ

0

∫

K−2h

∫h−Nρ

(x− y

h

)|wt(x)− wt(y)|dydxdt ≤

≤ 4ωK(h) + 4

∫ρ(σ)

∫ T

0

∫

K−2h

|uε(t, x)− uε(t, x− hσ)|dxdtdσ ≤ 4ωK(h) + 4ωK(h).

The function ωK(τ) = min0<h≤δ/2

CK

τhm + ωK(h) + ωK(h)

majorates I and tends to 0 as

τ → 0 , which ends the proof. ⋄

A3. The Minty-Browder argument

Lemma 7 Let E be a Banach space, E ′ its dual and (·, ·) denote the duality product of

elements of E ′ and E . Take a sequence vn in E such that vn v . Take a sequence

of monotone operators An : E 7→ E ′ such that An converge pointwise to some operator

A : E 7→ E ′ and Anvn∗ χ in E ′ .

Then χ = Av whenever A is hemicontinuous (i.e., continuous in the weak-∗ topology

of E ′ along each direction) and lim infn→∞(Anvn, vn) ≤ (χ, v) .


Proof of Lemma 7: The proof is standard (e.g., see [JLL]); we give it here for the sake

of completeness. Under the assumptions of Lemma 7, for all η ∈ E one has

(χ, v − η) ≥ lim infn→∞

(Anvn, vn − η) ≥ lim infn→∞

(Anη, vn − η) = (Aη, v − η).

Taking η = v + λζ with λ ∈ IR , ζ ∈ E and letting λ increase to zero, one gets (χ, ζ) ≥(Av, ζ) . As λ decreases to zero, the inverse inequality follows, so that (χ, ζ) = (Av, ζ) for

all ζ ∈ E . ⋄

CHAPTER 2.II

Convergence of Finite Volumes Approximations

for a Nonlinear Elliptic-Parabolic Problem:

a Variational Approach†

Introduction

Let Ω be an open bounded polygonal domain in Rd , d ≥ 1 and T > 0 . We consider the

initial boundary value-problem for a system of nonlinear elliptic-parabolic equations:

b(v)t = div ap(Dv) on Q = (0, T )× Ω,

v = 0 on Σ = (0, T )× ∂Ω,

b(v)(0, ·) = u0 on Ω,

(1)

where 1 < p <∞ and div ap(Dv) = div (|Dv|p−2Dv) is the N -dimensional p -Laplacian,

N ≥ 1 , i.e.,

ap : ξ = (ξ1, . . . , ξN) ∈ (Rd)N 7→ |ξ|p−2ξ =(∑i,j |ξji |2

)p/2−1

(ξ1, . . . , ξN) ∈ (Rd)N .(2)

We assume that

b : RN → RN is continuous, monotone with b(0) = 0, i.e.,

there exists a convex differentiable function Φ : RN → R

with Φ(0) = 0 such that b = ∇Φ,

(3)

and

u0 ∈ L1(Ω)N with Ψ(u0) ∈ L1(Ω), (4)

where Ψ is the Legendre transform of Φ given by

Ψ : z ∈ RN 7→ sup

σ∈RN

∫ 1

0

(z − b(sσ)) · σ ds.

†The results of this chapter are being prepared upon publication [BAGuW]

126 Variational Approach for a Finite Volume Method

Systems of elliptic-parabolic equations of type (1) arise as a model of flow of (several) fluids

through porous media (cf. e.g. [Bear, DiDT94]). They have already been studied extensively

in the literature in the last decade from a theoretical point of view (cf. e.g. [ALpr, AL83,

Kac90, DiDT94, BW96, Ot96, Bou97, BW99, CaW99, BAB]). Existence of weak solutions of

general systems of elliptic-parabolic equations has been proved in [ALpr, AL83], using Galerkin

approximations and time-discretization. Similar results have been obtained later by other

authors using different methods (e.g., using a semigroup approach as in [BW96, Bou97] in the

case N = 1 ).

In particular, it is known that in the case of the system (1), for any u0 satisfying (4),

there exists a weak solution of (1), where the weak solution is defined as follows.

Definition 1 (Weak solution) A function v ∈ E = Lp(0, T ;W 1,p0 (Ω))N is a weak solution

of the problem (1), if b(v) ∈ L∞(0, T ;L1(Ω))N , the function b(v)t ∈ E ′ = Lp′(0, T ;W−1,p′(Ω))N

(where p′ denotes the conjugate exponent of p ) satisfies

< b(v)t, φ >E′,E +

∫ ∫

Q

ap(Dv) ·Dφ = 0 (5)

for all φ ∈ E , where < ·, · >E′,E denotes the duality pairing between E ′ and E , and

− < b(v)t, ξ >E′,E=

∫ ∫

Q

b(v)ξt +

∫

Ω

u0ξ(0) (6)

for all ξ ∈ E with ξt ∈ L∞(Q)N and ξ(T ) = 0 .

Moreover, if v is a weak solution of (1), then, by the “chain rule” lemma of [AL83],

B(v) ∈ L∞(0, T ;L1(Ω))N , where

B : z ∈ RN 7→ b(z) · z − Φ(z) ≡

∫ 1

0

(b(z)− b(sz)) · zds ≡ Ψ(b(z)) ∈ R. (7)

From the results of [Ot96, CaW99] it also follows that, in the scalar case N = 1 , there is

uniqueness of a weak solution of (1). To our knowledge, the question of uniqueness is open

in the case N ≥ 2 .

The variational approach of [AL83] for elliptic-parabolic problems has been revisited in

Chapter 2.I. Beyond an extension of different existence results, in Chapter 2.I a concise variant

of the techniques of [AL83], applied to prove continuity of weak solutions of general elliptic-

parabolic problems with respect to the data and coefficients, has been presented. In this

chapter we are interested in proving, with the same techniques, the convergence of approxi-

mations by finite volumes numerical schemes for the model nonlinear elliptic-parabolic problem

(1).

Finite volumes methods are well suited for numerical simulation of processes where exten-

sive quantities are conserved, and it is a very popular method among engineers in hydrology

where systems of equations of this type arise. Therefore justification of convergence of this

2.II.0. Introduction 127

numerical approximation process is of particular interest. In [EGH98] the finite volume meth-

ods has been studied and convergence of this approximation procedure has been proved for

problem (1) in the particular case p = 2 , N = 1 . The same method has also been studied

for this equation (i.e. p = 2, N = 1 ) in the presence of an additional convection term (cf.

[EGH99]), and for a nonlinear diffusion problem in [EGHNS98].

In this chapter we deal with convergence for time implicit finite volume approximations of

(1). Let us emphasize that our main object is not only to prove the convergence of the finite

volume method for (1), but to develop the variational approach for this proof. The main idea of

this adaptation is to rewrite the discrete finite volume scheme under an equivalent continuous

form and to apply known stability techniques for the continuous equation (cf. [AL83, BAB])

in order to get convergence of the discrete approximation scheme.

In Section 1, we introduce the finite volume scheme for (1). We specify a class of admissible

partitions T h of Ω and (T h, kh) of Q (cf. Definitions 2 and 3) and define the approximate

discrete problems (Ph) ( h > 0 being the discretization parameter). To this end we have to

introduce a finite volume approximate Dh of the gradient operator D . A class of admissible

discrete gradient approximations is defined (see (14),(16) and Definition 4 below) and an

example of an admissible gradient approximation is given (cf. Remark 2).

In Section 1 we also state the main result of the chapter: for any admissible family of

grids T h, kh and gradient approximations Dh , there exists a sequence h→ 0 such that

solutions vh of the discrete problems (Ph) (cf. (24)-(26)) converge weakly in L1(Q)N to

a weak solution of (1).

In Section 2, existence of solutions vh of the discrete equations is established. Note that,

at least for special choices of the gradient approximation, it is also possible to prove uniqueness

of a discrete solution. We derive a discrete analog of Lp(0, T ;W 1,p0 (Ω))N a priori estimate

for vh and obtain a bound of B(vh) in L∞(0, T ;L1(Ω))N .

Section 3 is devoted to the problem of rewriting the discrete scheme under the form of an

equivalent equation

uht = div ap(Dvh) (8)

in D′(Q) for appropriate approximations uh ∈ L1(Q)N of b(vh) and Dv ∈ Lp(Q)N of

Dhvh .

In Section 4 we give the proof of the main theorem. The proof is essentially based on

three arguments: the a priori estimates obtained by using the chain rule argument of [AL83], a

lemma of Kruzhkov (cf. [K69a]) to get strong compactness in the parabolic part, and a Minty-

Browder argument (cf. e.g. [JLL]) to get convergence in the elliptic part of the continuous

form of the approximate equations.


Sections 5 and 6 are devoted to the proof of several auxiliary results and technical details

used in the proof of the main theorem. In particular, in Section 5, in addition to the contin-

uous approximation Dvh of the discrete gradients Dhvh defined in Section 3, we construct

auxiliary continuous approximations vh of the discrete solutions vh themselves. In this step,

it is convenient to impose the assumption (10) of proportionality of the mesh. Moreover, this

assumption is essentially used in the proof of the discrete Poincare inequality (cf. Lemma 9 in

the Appendix) when p > 2 .

Section 7 is devoted to the proof of consistency of the finite volume approximation of

the elliptic operator in (1) in the sense of Definition 5 (cf. Section 7). Here we develop an

additional series of arguments. The restriction (iv) of Definition 4 (cf. Section 1) is used at

this stage.

In the Appendix we have collected several auxiliary results form the theory of Sobolev

spaces and their discrete analogues.

Throughout the chapter we keep the notations introduced in this section. Moreover, as an

abuse of notation, in the following sections, the spaces Lp(Q)N , Lp(Ω)N etc. are written

as Lp(Q) , Lp(Ω) etc.

1 The numerical scheme

In order to construct approximate solutions to Problem (1), we use in this section the implicit

discretization in time and a finite volume scheme in space.

First we introduce a notion of admissible mesh of Ω (see also [EyGaHe] and [EGH99]).

Definition 2 (Admissible mesh) Let Ω be an open bounded polygonal subset of Rd . An

admissible finite volume mesh T of Ω is given by : a family of open polygonal convex

subsets with positive measure of Ω called ”control volumes” (for the sake of simplicity, we

shall denote by T the family of control volumes); a family E of subsets of Ω contained in

hyperplans of Rd , with positive (d−1) -measure (these are the edges of the control volumes);

a family of points of Ω , where these families satisfy the following properties:

(i) The closure of the union of all the control volumes is Ω ;

(ii) For any (K,L) ∈ (T )2 with K 6= L , either the length of K ∩ L is 0 or K ∩ L = σ

for some σ ∈ E . Then we will denote σ = K|L .

(iii) For any K ∈ T , there exists a subset EK of E such that ∂K = K \K = ∪σ∈EKσ .

Furthermore, E = ∪K∈T EK and we will denote by N (K) the set of boundary volumes

of K that is N (K) = L ∈ T , K|L ∈ EK .

(iv) The family of points (xK)K∈T is such that xK ∈ K (for all K ∈ T ) and, if

σ = K|L , it is assumed that the straight line (xK , xL) is orthogonal to σ .

2.II.1. Numerical scheme 129

In the sequel we will use the following notation. The size of the mesh (or space step)

is defined by: size(T ) := maxK∈T δ(K) , where δ(K) denotes the diameter of the control

volume K . For any K ∈ T and σ ∈ E , m(K) is the d -dimensional Lebesgue measure

of K and m(σ) the (d−1) -dimensional Lebesgue measure of σ . The set of adjacent

couples (K,L) is denoted by Υ . The set of interior (resp. boundary) edges is denoted by

Eint (resp. Eext ), that is Eint = σ ∈ E ; σ 6⊂ ∂Ω (resp. Eext = σ ∈ E ; σ ⊂ ∂Ω ).

The set of external control volumes is denoted by Text . For all K ∈ T , L ∈ N (K) and

σ ∈ EK , we denote by xσ the orthogonal projection of xK on σ (thanks to assumption

(iv), this orthogonal projection is the same from xK or xL if σ = K|L ), by dK,L (resp.

dK,σ ) the Euclidean distance between xK and xL (resp. xσ ) and by νK (resp. νK,L

and νK,σ ) the outside unit normal to K (resp. with respect to L and with respect to σ ).

Remark that νK |σ = νK,σ for all σ ∈ EK and that thanks to assumption (iv), νK,L = νK,σ

if σ = K|L . Finally for all K ∈ T and σ ∈ EK we denote by S(K, σ) (resp. S(σ) ) the

”half-diamond” (resp. ”diamond”) associated to K and σ (resp. to σ ) that is the smallest

convex set that contains σ and xK (resp. S(σ) = S(K, σ) ∪ S(L, σ) ).Further, let T be an admissible mesh in the sense of Definition 2. Together with k ∈

(0, T ) , it generates the space-time grid ((n− 1)k, nk)×KK∈T ,n=1,...,[T/k]+1 , which we

denote by (T , k) . We define the grid step h by h := maxk, size(T ) . We denote by

QnK the space-time volumes, i.e., Qn

K = ((n − 1)k, nk) ×K for all n = 1, . . . , [T/k] + 1

and all K ∈ T . The lateral boundary of QnK is denoted by Σn

K . In addition, we introduce

the notation ν = 1/2minK,σ∈N (K) dK,σ and denote by CnK the cartesian product of some

fixed d -dimensional cube inside K , with edge 2ν/√d and center xK , and the interval

((n − 1)k, nk) . Furthermore, we will use the notation Υκ(QnK) for the union of all space-

time volumes of (T , k) that are separated from QnK by at most κ ∈ N (space or time)

interfaces.

Definition 3 (Admissible families of meches and grids) A family of meshes T is ad-

missible if each mesh is admissible and the following assumptions hold with some fixed M, ζ :

(v) there exists M such that

M ≥ maxK∈T

card(EK); (9)

(vi) there exists ζ > 0 such that

ζ size(T ) ≤ minK∈T ,σ∈EK

dK,σ. (10)

A family of space-time grids (T h, kh) , parametrized with h ∈ (0, 1) , is admissible if the

family T h of space meshes is admissible and maxkh, size(T h) ≤ h . In the sequel, we

write k for kh and (T h, k) for (T h, kh) .


Remark 1 While Definition 2 and assumption (9) impose standard assumptions on the grid

(cf. [EyGaHe]), the hypothesis (10) is a strong proportionality condition. In fact, for p ≤ 2

it is only a technical assumption which permits to prove Lemma 5 (cf. Section 5). Lemma 5

is a pure qualitative result, in the sense that no estimation on the functions vh constructed

in this lemma is used in the sequel, but only the fact of their existence. Elsewhere, we can

replace (10) by the usual (cf. [EyGaHe]) assumption

ζ ≤ minK∈T h,σ∈EK

dK,σ

δ(K), (11)

which should be the only condition taken into account from the numerical point of view.

Nevertheless, in the case p > 2 the restriction (10) is essentially used in the proof of the

discret Poincare inequality (cf. the Appendix).

When using a finite volume method to approximate problem (1), we consider an ap-

proximate solution (if it exists) which is piecewise constant. Then we need to construct an

approximation of the gradient. Let us consider a set of values (vnK)K,n ⊂ RN (for the sake of

simplicity, we will make an abuse of notation by omitting K ∈ T h , n = 1, . . . , [T/k] + 1 )

and the dicrete solution defined by

vh|QnK= vnK . (12)

We construct the approximation of the gradient in the following way. The normal component

of the ”gradient” of the discrete solution vh is approximated by the operator Gh⊥ defined by

Gh⊥ : (vnK)K,n 7→ (gm⊥,σ)σ,m,

gm⊥,σ =

∣∣∣∣vmL − vmKdK,L

∣∣∣∣ ∈ [0,+∞) for σ = K|L,(13)

while the whole of the ”gradient” of the discrete solution vh on interfaces of the control

volumes is approximated by an operator Gh defined by

Gh : (vnK)K,n 7→ (gmσ )σ,m,

gmσ ∈ (Rd)N for all σ ∈ E , m = 1, . . . , [T/k] + 1.(14)

Then we extend Gh to the whole of Q by the lifting operator Lh defined by

Lh : (gmσ )σ,m 7→ Lh ((gmσ )σ,m) ∈ Lp(Q),

Lh ((gmσ )σ,m) |((m−1)k,mk)×S(σ) = gmσ for all σ ∈ E , m = 1, . . . , [T/k] + 1.(15)

We define the discrete gradient operator Dh by

Dh : (vnK)K,n 7→ (Lh Gh)((vnK)K,n) ∈ Lp(Q). (16)

2.II.1. Numerical scheme 131

It is convenient to extend the gradient approximation procedure to functions in E . Let

introduce the averaging operator Mh defined on E = Lp(0, T ;W 1,p(Ω))N by

Mh : η 7→ Mh η = (ηnK)K,n, ηnK =1

|CnK |

∫∫

CnK

η; (17)

Then by an abuse of notation, we also write Dh for the operator

Dh : η ∈ E 7→ (Lh Gh Mh)(η) ∈ Lp(Q), (18)

and Dh⊥ for the operator

Dh⊥ : η ∈ E 7→ (Lh Gh

⊥ Mh)(η) ∈ Lp(Q). (19)

Taking in (17) averages over cylinders CnK is convenient for expressing the consistency of the

gradient approximation for affine in x functions (cf. (iv) in the definition below). The crucial

property is the symmetry with respect to the axis x = xK .

Definition 4 (Admissible family of discrete gradient operators) For a given family of

meshes T h , a family of corresponding discrete gradient operators Dh is admissible, if

Dh and the corresponding operators Gh have the following properties:

(i) For each h , Gh is linear, i.e.,

Gh((vnK + wnK)K,n) = Gh((vnK)K,n) + Gh((wn

K)K,n) (20)

(ii) For each h , Gh is consistent with Gh⊥ , i.e.,

gmK|L νK,L = sign(vmL − vmK ) gm⊥,K|L for all (K|L) ∈ Υ, m = 1, . . . , [T/k] + 1. (21)

(iii) The family Dh is uniformly local, i.e., there exists κ ∈ N independent of h such that

for all K ∈ T h , all n = 1, . . . , [T/k] + 1 and all set of values (vnK)K,n of RN , there

exists a constant C which only depends on p , d and M , ζ in (R2) , (R3) such

that

∥∥Dh((vnK)K,n)∥∥Lp(Qn

K)≤ C

∥∥Dh⊥((v

nK)K,n)

∥∥Lp(Υκ(Qn

K)). (22)

(iv) For each h , Gh is consistent with affine functions. More exactly, assume that, for

K ∈ T h and n ∈ N given, there exists a constant c ∈ (Rd)N and a function w ∈ E

such that Dw ≡ c on Υκ(QnK) and that vlL = 1

|CnK |

∫∫CnKw whenever Ql

L ⊂ Υκ(QnK) .

Then

gnσ = c for all σ ∈ EK . (23)


Below we provide an example of gradient approximation that complies with the properties

(i)-(iv) above. For simplicity, we restrict our attention to the 2D case.

Remark 2 (Example of admissible discrete gradients in 2D) Let T h be a family of

admissible meshes of Ω in the sense of Definition 2. For T ∈ T h , for each σ0 ∈ E , let

σ1 , σ2 , σ3 and σ4 be the four adjacent edges (see figure (1)). Let x1 (resp. x2 ) be

the intersection point of σ0 with σ1 and σ2 (resp. σ3 and σ4 ). Then we construct the

approximate gradient on σ0 in the following way.

We take the standard values gn⊥,σ0, gn⊥,σ1

of the normal components of the discrete

gradient on σ0 and σ1 , respectively. The edges are not colinear, so that there exists a

unique vector that has these values as projections on the normal directions to σ0 and σ1 ,

respectively. We draw this vector in x1 and denote it by gnσ01. In the same way, we can

reconstruct gnσ02(resp. gnσ03

, gnσ04) in x1 (resp. x2 ) from the normal components of the

gradients on σ0 and σ2 (resp. σ3 , σ4 ).

σ0

σ1

σ2

σ3

σ4

gn⊥,σ0

gn⊥,σ1

x1

x2

gnσ01

gnσ02

gnσ03

gnσ04

gnσ0

Figure 1: Reconstruction of gradient on σ0

Then the approximate gradient on σ0 (or in S(σ0) , according to (16)) is given by

gnσ0=

1

4(gnσ01

+ gnσ02+ gnσ03

+ gnσ04).

It is easy to show that, if the family T h is admissible in the sense of Definition 3, this

approximation of the gradient is admissible in the sense of Definition 4.

2.II.2. Existence of a discrete solution and a priori estimates 133

We are now in order to write the scheme. The equation for the scheme is given by

m(K)b(vnK)− b(vn−1

K )

k=

∑

L∈N (K)

m(K|L) ap(gnK|L) νK,L for all K ∈ T , n ∈ N. (24)

The initial condition is given by some values

u0K = b(v0K) for all K ∈ T , (25)

and the homogeneous Dirichlet boundary condition is taken into account in the following way:

vnK = 0 for all K ∈ Text, n ∈ N. (26)

The discrete Problem (Ph) corresponding to a grid (T h, k) , where T h is an admissible

mesh of Ω in the sense of Definition 2, is given by the discrete equation (24), the initial

condition (25) and the boundary condition (26). In Section 2 below, we prove that there

exists a solution vh to the discrete Problem (Ph) .

The values (u0K)K will be chosen in order to comply, at the limit h→ 0 , with the initial

condition in (1) and the restriction (4) on it, i.e.,u0,h → u0 in L1(Ω)N as h→ 0,

Ψ(u0,h) → Ψ(u0) in L1(Ω) as h→ 0,(27)

where u0,h|K = u0K . Our main result is the following theorem.

Theorem 1 (Convergence) Let (T hm , khm)m∈N be a sequence of admissible grids in the

sense of Definition 3 such that hm = maxkhm, size(T hm)

< 1 , hm → 0 as m→ +∞ .

Assume that (3),(4), and the analogue of (27) hold. For each m ∈ N , let (vhm)m∈N

be a discrete solution to the problem (Phm)m∈N , where (Dhm)m∈N is an admissible family

of discrete gradient operators in the sense of Definition 4. Then there exists a subsequence

(hmk)k∈N , hmk

→ 0 as k → ∞ , such that vhmk v in L1(Q) as hmk→ 0 , where

v ∈ E = Lp(0, T ;W 1,p0 (Ω))N is a weak solution of the problem (1) in the sense of Definition 1.

In the sequel, we will omit subscripts in sequences (hm) , (hmk) .

2 Existence of a discrete solution and a priori estimates

We will repeatedly use the following remark.

Remark 3 (Discrete integration by parts) Let T be an admissible mesh of Ω in the

sense of Definition 2. Let (vK)K∈T ⊂ RN and (FK,L)(K,L)∈Υ ⊂ R

N be real values such

that vK = 0 for all K ∈ Text and FK,L = −FL,K for all (K,L) ∈ Υ . Then

∑

K∈T

vK∑

L∈N (K)

FK,L =∑

(K,L)∈Υ

(vK − vL)FK,L. (28)


Now we can state the result for existence of a discrete solution.

Theorem 2 (Existence) Let (T , k) be a space-time grid, where T is an admissible mesh

of Ω in the sense of Definition 2. Let Dh be a discrete gradient operator having the properties

(i)-(iii) of Definition 4. Assume that (3) holds; then there exists a solution vh to the discrete

problem (Ph) .

Remark 4 (Uniqueness) While uniqueness of a waek solution of the problem (1) itself for

N ≥ 2 seem to be an open problem, for special choices of the gradient approximation we can

also prove the uniqueness of a solution to the discrete problem (Ph) .

Proof of Theorem 2: Fix n ∈ 1, . . . , [T/k]+1 . Assume that the values (vn−1K )K∈T

are already found. We denote by V the vector of (RN)cardT whose entries (vnK)K∈T satisfy

the condition (26). Let us consider the operator S that associates to a “vector” V the

“vector” given by (24), i.e.,

S(V) =

m(K)

b(vnK)− b(vn−1K )

k−

∑

L∈N (K)

m(K|L) ap(gnK|L) · νK,L

K∈T

.

We are looking for a solution to the equation S(V) = 0 . Consider the scalar product

(S(V),V) in (RN)cardT . We have

1k

∑

K∈T

m(K) b(vnK) · vnK − 1k

∑

K∈T

m(K) b(vn−1K ) · vnK −

−∑

K∈T

∑

L∈N (K)

m(K|L) vnK ap(gnK|L)νK,L = 0.

(29)

In view of hypothesis (3), we have that the first term on the left-hand side of (29) is non-

negative. Since all the norms are equivalent on (RN)cardT , for the second term on the

left-hand side of (29) we have

1

k

∑

K∈T

m(K) b(vn−1K ) · vnK ≤ |V| 1

k

∑

K∈T

m(K) b(vn−1K ) = C |V| ;

here and in the sequel of the proof, C denotes a positive constant independent of V , and

|V| denotes the euclidean norm of V . We then handle the last term on the left-hand side of

(29), which we denote by AV . Using the discrete integration by parts (Remark 3), we obtain

that

−AV =∑

(K,L)∈Υ

m(K|L) (vnL − vnK) ap(gnK|L)νK,L.

In view of the definition (2) of ap and thanks to hypothesis (21), we obtain

−AV =∑

(K,L)∈Υ

m(K|L)∣∣gnK|L

∣∣p−2 (vnL − vnK)2

dK,L. (30)

2.II.2. Existence of a discrete solution and a priori estimates 135

For p ≥ 2 , we have in view of (21) that

∣∣gnK|L

∣∣p−2 ≥∣∣gn⊥,K|L

∣∣p−2=

∣∣∣∣vnL − vnKdK,L

∣∣∣∣p−2

,

which together with (30) yields

−AV ≥∑

(K,L)∈Υ

m(K|L) dK,L


∣∣∣∣p

.

Thanks to the discrete Poincare inequality (cf. Lemma 9 in the Appendix), we finally obtain

−AV ≥ α∑

K∈T

m(K) |vnK |p ≥ C |V|p ,

thanks to the equivalence of the norms on (RN)cardT .

For 1 0 such that, for all

(K,L) ∈ Υ ,

C∣∣gnK|L

∣∣ ≤ |V| .

Together with (30), this yields

−AV ≥ C |V|p−2∑

(K,L)∈Υ

m(K|L) dK,L


∣∣∣∣2

.

Thanks to the discrete Poincare inequality (cf. Lemma 9 in the Appendix) for p = 2 , we

finally obtain

−AV ≥ C |V|p−2∑

K∈T

m(K) |vnK |2 ≥ c |V|p

with some constant c > 0 independent of V , thanks to the equivalence of the norms on

(RN)cardT .

Returning to (29), we obtain for all 1 0 such

that

(S(V),V) ≥ c |V|p − C |V| ≥ 0,

for |V| large enough. Therefore in view of the Brouwer fixed point theorem (e.g., cf. [JLL,

Lemme 4.3]), there exists a solution to S(V) = 0 , i.e., there exists a solution to (24) with

condition (26) for (vn−1K )K given. ⋄

Proposition 1 Let(T h, k)

be an admissible family of grids in the sense of Definition 3.

Let Dh be a family of discrete gradient operators satisfying (ii) and (iii) of Definition 4.

Assume that (3),(4), and (27) hold. Then for all family (vnK)K,n of solutions of the discrete

problem (Ph) there exists a constant C which only depends on p , Ω , T and ‖Ψ(u0)‖L1(Ω)

such that∫∫

Q

∣∣Dhvh∣∣p ≤ C, (31)

and

∑

K∈T h

m(K)B(vnK) ≤ C, for all n = 1, . . . , [T/k] + 1. (32)

Proof : Take i ∈ 1, . . . , [T/k] + 1 and multiply each term in (24) by viK . By (26),

using the discrete integration by parts (Remark 3) and (21), one gets

∑

K∈T h

m(K)(b(viK)− b(vi−1K )) · viK +

+k∑

(K,L)∈Υ

m(K|L)∣∣∣giK|L

∣∣∣p−2 ∣∣∣gi⊥,K|L

∣∣∣ |viK − viL| = 0.

By the convexity of Φ , one has (b(viK)−b(vi−1K )) ·viK ≥ B(viK)−B(vi−1

K ) . Summation over

i from 1 to n ∈ 1, . . . , [T/k]+1 , and recalling the definition of the discrete gradient and

its normal component, we infer

∑

K∈T h

m(K)B(vnK) + d

∫ nk

0

∫

Ω

∣∣∣Dhvh∣∣∣p−2∣∣∣Dh

⊥vh∣∣∣2

≤∑

K∈T h

m(K)Ψ(u0K). (33)

Note that, by (21), for 1 < p < 2 the value∣∣∣Dhvh

∣∣∣p−2∣∣∣Dh

⊥vh∣∣∣2

can be set zero whenever

Dh⊥v

h is zero. Therefore the integral in (33) always makes sense.

Now (32) follows directly from (27) and (33). As to (31), there are two cases. For p ≥ 2 ,

one has by (21) and (33)∫ ∫

Q

∣∣∣Dh⊥v

h∣∣∣p

≤∫ ∫

Q

∣∣∣Dhvh∣∣∣p−2∣∣∣Dh

⊥vh∣∣∣2

≤ const.

For 1 < p < 2 , using the “inverse” Holder inequality with the exponents p/2 < 1 and

p/(p− 2) < 0 , one gets

(∫ ∫

Q

∣∣∣Dh⊥v

h∣∣∣p)2/p

≤(∫ ∫

Q

∣∣∣Dhvh∣∣∣p)(2−p)/p ∫ ∫

Q

∣∣∣Dhvh∣∣∣p−2∣∣∣Dh

⊥vh∣∣∣2

.

In the two cases, it follows by (22) that∫∫

Q

∣∣∣Dh⊥v

h∣∣∣p

≤ const independently of h . Using

(22) again, one obtains (31). ⋄

2.II.3. Rewriting discrete equations under continuous form 137

3 Rewriting discrete equations under continuous form

Let vh be the discrete function (12) produced by the finite volume scheme (24)-(26) on a grid(T h, k

)of size h . We will replace the discrete gradient Dhvh by a function Dvh ∈ Lp(Q)

and the function b(vh) by a function uh ∈ L1(Q) so that (24) is equivalent to (8), i.e.,

uht = div ap(Dvh)

in D′(Q) . This representation plays the key role in proving the convergence result of Theo-

rem 1.

Define uh as the piecewise affine in t approximation of b(vh) :

uh(t, x)|QnK= b(vnK) +

t− kn

k(b(vnK)− b(vn−1

K )) (34)

Besides, for given K, n and a set(An

σ

)σ∈EK

⊂ (Rd)N , let AnK be defined a.e. on ∂K by

AnK |σ = An

σ for all σ ∈ EK . Let νK be the exterior unit normal vector to ∂K . Consider

the following Neumann problem in the factor space W =W 1,p(K)/R :

div ap(Dw) =1

m(K)

∫

∂K

AnKνK on K

ap(Dw)νK|∂K = AnKνK .

(35)

Lemma 1 Let AnK ∈ Lp′(∂K) . Then there exists a unique distribution solution wn

K to (35)

in W . This solution gives the global minimum to the functional

L : w ∈ W 7→ Lw =1

p‖Dw‖pLp(K) −

∫

∂K

wAnKνK +

1

m(K)

∫

K

w

∫

∂K

AnKνK . (36)

Proof : Supply W with the norm ‖w‖W =(∫

K|Dw|p

)1/p(we will ambiguously denote

by the same symbol an element of W 1,p(K) and the corresponding equivalence class as

element of W ). Consider the operator A : w ∈ W 7→ Aw ∈ W ′ defined by

< Aw,ϕ >W ′,W=

∫

K

ap(Dw) : Dϕ (37)

for all ϕ ∈ W , and the functional f ∈ W ′ defined by

< f, ϕ >W ′,W= − 1

m(K)

∫

K

ϕ

∫

∂K

AnKνK +

∫

∂K

ϕAnKνK (38)

for all ϕ ∈ W . Note that f is well defined, since the right-hand side of (38) is invariant

under translation by a constant in W 1,p(K) .

The operator A in (37) is bounded, hemicontinuous, strictly monotone on W , and

< Aw,w >W ′,W

‖w‖W=

∫K|Dw|p

‖w‖W= ‖w‖p−1

W → ∞


as ‖w‖W → ∞ . Thus A is bijective (cf. e.g. [JLL, Chapitre 2,Theoreme 2.1]).

Besides, the functional L is well defined, convex on W , and

Lw ≥ 1

p‖Dw‖pLp(K) − ‖An

K‖Lp′(∂K)

∥∥∥∥w − 1

m(K)

∫

K

w

∥∥∥∥Lp(∂K)

≥

≥ 1

p‖Dw‖pLp(K) − const‖Dw‖Lp(K)

by Lemmae 12,11 (cf. the Appendix). Hence Lw → +∞ as ‖w‖W → ∞ , so that L

attains its global minimum on W (cf. e.g. [Br, Corollaire III.20]). Using the relation of the

p-laplacian with the Lp norm, by the standard variational argument we find out that this

minimum is attained at the unique solution wnK of the equation Aw = f in W ′ . ⋄

Let vnK be the solution of (35) with

AnK = ap(g

nσ) for σ ∈ EK , (39)

where gnσ are taken from (14), and vnK is normalized by assigning

1

m(K)

∫

K

vnK = vnK . (40)

We introduce the discrete-continuous approximation of Dhvh by setting

Dvh(t, x)|QnK= DvnK(t, x). (41)

Let us also define vh by

vh(t, x)|QnK= vnK(t, x). (42)

Clearly, we have uht = div ap(Dvh) pointwise on Qn

K . Note that Dvh is only the pointwise

gradient of vh , while the gradient of vh in the sense of distributions contains Dirac masses

concentrated on grid edges. Nevertheless, one has the following result.

Proposition 2 (The continuous form of (24)) Assume that (vnK)K,n verifies (24). Let

uh and Dvh be defined by (34) and (41), respectively. Then uht = div ap(Dvh) holds in

D′(Q) .

Proof : By the local conservativity of the scheme, the fluxes are continuous on the

space interfaces: AnK|L(x) = An

L|K(x) for all x ∈ K|L , for all (K,L) ∈ Υ , all n ∈

2.II.4. Proof of Theorem 1 139

1, . . . , [T/k] + 1 . Moreover, uh is continuous on the time interfaces. Therefore, for all

ph ∈ D(Q) ,

∫∫

Q

(uh · ϕt − ap(Dvh) : Dϕ) =

∑

K,n

∫∫

QnK

(uh · ϕt − ap(Dvh) : Dϕ) =

=∑

K,n

−∫∫

QnK

(uht − div ap(Dvh)) · ϕ+

∑

L∈N (K)

∫ kn

k(n−1)

∫

K|L

ϕAnK|LνK,L+

+

∫

K

ϕ(kn, ·) · uh(kn, ·)−∫

K

ϕ(k(n− 1), ·) · uh(k(n− 1), ·)=

=∑

n

∑

(K,L)∈Υ

∫ kn

k(n−1)

∫

K|L

ϕAnK|L(νK,L + νL,K)+

+

∫

K

ϕ(T, ·) · uh(T, ·)−∫

K

ϕ(0, ·) · uh(0, ·) = 0,

so that (8) holds. ⋄

In addition to this result, it will be useful to have in hand some “Lp(0, T ;W 1,p0 (Ω)) -

version” of vh for each h . We will call vh a family of continuous in x approximations

of vh , if

vh ∈ E with ‖vh‖E ≤ const‖Dhvh‖Lp(Q) (43)

with a constant independent of h ,

‖vh − vh‖L1(Q) → 0 as h→ 0, (44)

and

1

km(K)

∫∫

QnK

vh = vnK =1

|CnK |

∫∫

CnK

vh. (45)

The existence of such approximations is proved in Lemma 5 (cf. Section 5) for the case where

(10) holds.

4 The proof of Theorem 1

In Chapter 2.I, in the context of continuous dependence upon the data of weak solutions to

“general” elliptic-parabolic problems, the convergence proof for weak solutions of approximat-

ing problems has been reduced to the three essential arguments:

(A) a priori estimates, by the chain rule argument of Alt-Luckhaus (cf. [AL83, Ot96,

CaW99, BAB]);

(B) strong compactness in the parabolic term, by the Kruzhkov lemma (cf. [K69a] and

Chapter 2.I, Lemma 6);


(C) convergence in the elliptic term, by the Minty-Browder argument (cf. e.g. [Mi62, Mi63,

Bro63, JLL] and Chapter 2.I, Lemma 7).

Here we will take advantage of the “continuous” form (8) of the system (24) and pass to the

limit in (a subsequence of) vh as h→ 0 by applying the same arguments to, respectively,

(A) vh , the continuous in x approximations of vh ;

(B) uh , the piecewise affine in t approximations of b(vh) ;

(C) Dvh , the discrete-continuous approximations of Dhvh .

Proof of Theorem 1: We will repeatedly refer to results contained in Sections 5-7

below. The proof consists of three steps.

(A) Let vh be a family of solutions to the family of discrete problems (Ph) . By Lemma 5

(cf. Section 5) there exist vh ∈ E satisfying (43)-(45). In particular, by Proposition 1,

‖vh‖E ≤ const uniformly in h . Hence there exists a subsequence h → 0 and a function

v ∈ E such that vh v in E as h → 0 . By (44), one also has vh v ∈ L1(Q) (cf.

also Remark 5).

(B) We claim that the family uh is relatively compact in L1(Q) . Indeed, extend uh

by zero on (R× Rd) \Q . Let us check the following three conditions:

(i) uh is bounded in L1(Q) ;

(ii) ap(Dvh) is bounded in L1(Q) ;

(iii) for all ∆ > 0 small enough, one has

sup|∆x|≤∆

∫∫

Q

|uh(t, x+∆x)− uh(t, x)| dxdt ≤ ωx(∆) (46)

uniformly in h , where ωx(∆) → 0 as ∆ → 0 .

In order to prove (i), note that

|b(z)| ≤ δB(z) + sup|ζ|≤1/δ

|b(ζ)| (47)

holds for all δ > 0 (cf. e.g. [AL83]). Hence we have by (34),(25)

‖uh‖L1(Q) ≤ 2

∫∫

Q

|b(vh)|+ k∑

K∈T h

m(K)|u0K| ≤∫∫

Q

B(vh) + const + k∑

K∈T h

m(K)|u0K |,

which is bounded uniformly in h , by Proposition 1 and (27).

As to (ii), one has

‖ap(Dvh)‖L1(Q) =

∫∫

Q

|Dvh|p−1 ≤ ‖Dvh‖p−1Lp(Q)|Q|1/p ≤ const

2.II.4. Proof of Theorem 1 141

by the a priori estimate on ‖Dvh‖Lp(Q) , which is proved in Lemma 2 (cf. Section 5).

The estimate (46) of (iii) is proved in Lemma 6, as a consequence of the a priori estimate

(31) (cf. Section 6).

Now we may conclude by Lemma 6 from Chapter 2.I that

sup|∆t|≤∆

∫∫

Q

|uh(t+∆t, x)− uh(t, x)| dxdt ≤ ωt(∆) (48)

uniformly in h , where ωt(∆) → 0 as ∆ → 0 . Thus there exists a subsequence h→ 0 and

a function u ∈ L1(Q) such that uh → u in L1(Q) and a.e. on Q . Besides, we have to

establish that u = b(v) , where v is the weak limit of vh in E . The proof, which follows

the idea of [BrSt73], is given in Lemma 7 (cf. Section 6).

(C) First note that uh → b(v) in L1(Q) , so that uht → b(v)t in D′(Q) . Moreover, by

(8) ‖uht ‖E′ = ‖ap(Dvh)‖Lp′(Q) = ‖Dvh‖p−1Lp(Q) , which is bounded by Lemma 2 (cf. Section 5).

Therefore uht is weak- ∗ relatively compact in E ′ .

It follows that there exists a subsequence, which we abusively denote by h→ 0 , such that

(i) vh v in E ;

(ii) −uht∗ −b(v)t in E ′ .

Moreover, for all h one has

(iii) uht = Ahvh , where vh is constructed in Lemma 5 and Ah is the operator that maps

η ∈ E to −divAhη ∈ E ′ , with Ah : E 7→ Lp′(Q) defined in (51) below.

Indeed, let us define the finite volume approximate Ah· of ap(D·) . For η ∈ E and all

space-time volume QnK , set

Dηh(t, x)|QnK= DηnK(x), (49)

where ηnK is the unique solution to the problem (35) with AnK |σ = ap

(((Gh Mh)η)nσ

)for

all σ ∈ EK . It is convenient to normalize ηnK by assigning

1

m(K)

∫

K

ηnK =1

km(K)

∫∫

QnK

η. (50)

Here Mh and Gh are the averaging operator and the gradient approximation operator,

respectively, defined by (17) and (14), respectively. Assign

Ahη = ap(Dηh). (51)

From (45) we have(Gh Mh

)vh = Gh

((vnK)K,n

). Therefore, Ahvh = ap(Dv

h) with Dvh

defined in (41), so that (8) yields (iii).

Arguing as in the proof of Proposition 2, one gets

< Ahη, ϕ >E′,E=

∫∫

Q

ap(Dηh) : Dϕ =

∫∫

Q

Ahη : Dϕ (52)

for all ϕ ∈ E . In particular, it follows that Ah is monotone for all h .

Now we apply Lemma 7 from Chapter 2.I to Ah and A : E 7→ E ′ defined by Aη =

−div ap(Dη) , or, equivalently, by

< Aη, ϕ >E′,E=

∫∫

Q

ap(Dη) : Dϕ (53)

for all ϕ ∈ E . Note that A is hemicontinuous. Two more assumptions of Lemma 7

(Chapter 2.I) have to be checked:

(iv) lim infh→0 < −uht , vh >E′,E ≤ < −b(v)t, v >E′,E ;

(v) for all η ∈ E , Ahη → Aη in E ′ .

In fact the property (v) expresses the consistency in E ′ of the finite volume approximation

of the operator −div ap(D·) on the space E . We prove in Theorem 3 (cf. Section 7) that

(v) actually holds under the assumptions of Theorem 1.

Before proving (iv), let us show that the initial condition (6) holds. From (34) we have

< −uht , ζ >E′,E= −∫∫

Q

uht · ζ =∑

K∈T h

m(K)u0K · ζ(0, ·) +∫∫

Q

uh · ζt (54)

for all ζ ∈ E with ζt ∈ L∞(Q) and ζ(T, ·) = 0 . By (27), we can pass to the limit in (54)

and obtain (6). Now we can apply the usual chain rule argument (cf. [AL83, Lemma 1.5])

and deduce that B(v) ∈ L∞(0, T ;L1(Ω)) and

< −b(v)t, v >E′,E= −∫

Ω

B(v)(T ) +

∫

Ω

Ψ(u0). (55)

(without loss of generality, we assume T to be a Lebesgue point of ‖B(v)‖L1(Ω)(·) ). Besides,by (8),(34),(45), and the monotonicity of b(·) , one has

< −uht , vh >E′,E= −1

k

∑

K,n

(b(vnK)− b(vn−1K )) ·

∫∫

QnK

vh =

=∑

K∈T h

[T/k]+1∑

n=1

m(K)(b(vnK)− b(vn−1K )) · vnK ≤

−∑

K∈T h

m(K)B(v[T/k]+1K ) +

∑

K∈T h

m(K)Ψ(u0K).

Together with (55),(27) and the Fatou lemma, this yields (iv).

We are now in position to conclude that −b(v)t = Av in E ′ , so that (5) also holds.

Thus v is a solution of (1) in the sense of Definition 1. ⋄

Remark 5 In fact, one could replace in the requirement (44) the space L1(Q) by the space

Lp(Q) , and show that vh v in Lp(Q) .

2.II.5. Two kinds of continuous approximations 143

5 Two kinds of continuous approximations

In this section we prove two auxiliary results concerning the approximations Dvh and vh of

Dhvh and vh , respectively. We also establish a uniform estimate on the space translates of

vh in Lq(Q) , 1 ≤ q ≤ p .

Lemma 2 Let(T h, k)

be an admissible family of grids, and (vnK)K,n be a solution of

(Ph) . Assume that the family of discrete gradient operators Dh satisfy (ii),(iii) of Defini-

tion 4, and let Dvh be defined by (41). Then ‖Dvh‖Lp(Q) ≤ const uniformly in h .

Proof : For all K, n fixed, the function vnK ∈ W 1,p(K) satisfies (35) in D′(K) . Take

vnK for the test function; it follows by (39) that∫

K

|DvnK |p =∑

σ∈EK

m(σ)( 1

m(σ)

∫

σ

vnK − 1

m(K)

∫

K

vnK

)(ap(g

nσ)νK,σ

).

Multiplying and dividing each term of the sum in the right-hand side by dK,σ/d , integrating

in t over (k(n− 1), kn) and summing over K, n , we obtain by the Holder inequality

‖Dvh‖pLp(Q) ≤ d(∑

K,n

∑

σ∈EK

1

dkm(K)dK,σ|ap(gnσ)|p

′)1/p′

×

×(∑

K,n

∑

σ∈EK

1

dkm(K)dK,σ

∣∣∣∣∣

1m(σ)

∫σvnK − 1

m(K)

∫KvnK

dK,σ

∣∣∣∣∣

p)1/p.

(56)

According to (18),(15) and (45), the first term in the right-hand side of (56) equals(∫∫

Q

(|Dhvh|p−1)p′)1/p′

= ‖Dhvh‖p−1Lp(Q) = ‖Dhvh‖p−1

Lp(Q),

which is bounded by Proposition 1. Besides, by Lemma 10 (cf. the Appendix) the second

term in the right-hand side of (56) is estimated by the value

const(∑

K,n

‖DvnK‖pLp(QnK)

)1/p= const‖DvnK‖Lp(Q).

Hence (56) yields

‖Dvh‖pLp(Q) ≤ const‖Dvh‖Lp(Q)

with a constant independent of h , which completes the proof. ⋄

Lemma 3 Let(T h, k)

be an admissible family of grids, (vnK)K,n be a solution of (Ph) ,

and vh be defined by (12) on Q and extended by zero on (R+×R) \Q . For x,∆x ∈ Rd ,

let Bx,x+∆x be the broken line that joins the centers of successive space mesh volumes crossed

by [x, x+∆x] ∩ Ω , and lx,x+∆x be the length of Bx,x+∆x ∩ Ω . Then for 1 ≤ q ≤ p ,∫∫

Q

|vh(t, x+∆x)− vh(t, x)|q dxdt ≤ ω(q)x (|∆x|) (57)

uniformly in h , with ω(q)x : R+ 7→ R

+ such that ω(q)x (∆) → 0 as ∆ → 0 . Moreover, one

can take ω(q)x (∆) = const∆(l(∆))q−1 , where l(∆) = supx∈Rd sup∆x∈Rd,|∆x|≤∆ lx,x+∆x .


Proof : Take x,∆x ∈ Rd and let Bx,x+∆x =

⋃(K,L)∈Υ | [x,x+∆x]∩(K|L)6=Ø[xK , xL] , where

xK , xL are the centers of the volumes K,L , respectively. Since

lx,x+∆x =∑

(K,L)∈Υ | [x,x+∆x]∩(K|L)6=Ø

dK,L,

convexity of the mapping y ∈ Rd 7→ |y|q yields

|vh(t, x+∆x)− vh(t, x)|q ≤ (lx,x+∆x)q−1

∑

(K,L)∈Υ | [x,x+∆x]∩(K|L)6=Ø

dK,L

∣∣∣∣vnK − vnLdK,L

∣∣∣∣q

for t ∈ (k(n − 1), kn) . For (K,L) ∈ Υ fixed, denote by ΩKL(∆x) the set of x ∈ Rd

such that [x, x+∆x] ∩ (K|L) 6= Ø . Clearly, ΩKL(∆x) is a prism of measure less or equal

than |∆x|m(K|L) . Hence∫∫

Q

|vh(t, x+∆x)− vh(t, x)|q ≤ (l(|∆x|))q−1∑

K,n

kdK,L

∣∣∣∣vnK − vnLdK,L

∣∣∣∣q

|ΩKL(∆x)| ≤

≤ d|∆x|(l(|∆x|))q−1

∫∫

Q

|Dh⊥v

h|q ≤ const|∆x|(l(|∆x|))q−1

by Proposition 1 and the Holder inequality. ⋄

For q > 1 , this result will be used together with the simple lemma below.

Lemma 4 Let T h be a family of admissible meshes of Ω ⊂ Rd parametrized by h =

size(T h) such that (10) holds. Let x,∆x ∈ Rd and Bx,x+∆x , lx,x+∆x , l(·) be defined as

in Lemma 3. Then l(∆) ≤ const (∆ + 2h) , where const only depends on d and on ζ in

(10).

Proof : Let Cx,x+∆x be the cylinder in Rd of radius h , with the segment [x−h ∆x

|∆x| , x+

∆x + h ∆x|∆x| ] for the axis. If the segment [x, x + ∆x] crosses a control volume K ∈ T h ,

K is contained in Cx,x+∆x . Note that |Cx,x+∆x| = const (|∆x| + 2h) hd−1 . On the other

hand, for all K ∈ T h , |K| ≥ const (ζh)d , and the maximum length of Bx,x+∆x ∩ K is

2h . Therefore l(|∆x|) is estimated by 2h times the number of control volumes contained

in Cx,x+∆x , i.e., l(|∆x|) ≤ 2h const (|∆x|+2h)hd−1

consthd , which concludes the proof. ⋄

Remark 6 In fact, we prove in Lemma 3 that for any mesh T admissible in the sense of

Definition 2, for all (wK)K ⊂ RN given, and w defined by w|K = wK , the norm of

w(·+∆x)−w(·) in Lq(Ω) is estimated by dC |∆x|(l(|∆x|))q−1 , where C is the discrete

W 1,q0 -norm of w . As it is shown in Lemma 4, for a family T h this estimate can be

improved to const dC |∆x|(|∆x| + 2h)q−1 , with const independent of h = size(T h) ,

provided one imposes the restriction of proportionality (10) on T h .

Moreover, calculating more carefully |ΩKL(∆x)| (cf. e.g. [EyGaHe, EGH98]), we can

prove this last estimate in case 1 ≤ q ≤ 2 without the restriction (10).

2.II.5. Two kinds of continuous approximations 145

Lemma 5 Let(T h, k)


and vh be defined by (12). Then there exists a family vh ⊂ E such that (43)-(45) hold.

Proof : We first convolute vh in x with a special mollifier, and then restore the average

over each mesh volume.

Let ν = 1/2minK∈T h,σ∈EK dK,σ . By (10) we have

h/ν ≤ const. (58)

Take ρ : x ∈ Rd 7→ d

|B(0,1)|dist (x, S(0, 1)) , where S(0, r) , B(0, r) are the (d− 1) -

dimensional sphere and the d -dimensional ball of centre 0 and radius r , respectively. Let

Ων = x ∈ Ω | dist (x, ∂Ω) > ν and χΩν be the characteristic function of Ων . We set

vh,ν =1

νdρ(xν

)∗ (vh χΩν ) (59)

and construct vh as

vh = vh,ν +∑

K∈T h

αKϕK . (60)

Here ϕK(t, x) = ϕK(x) = m(K)νd

π(x−xK

ν

). Here π : R

d 7→ R is a function with the

properties supp π = x ∈ Rd | 1 ≤ |x| ≤ 2 , π ≥ 0 ,

∫π = 1 , π ∈ C∞(Rd) ; xK is the

center of K , and

αK(t, x)|QnK= αn

K = vnK − 1

km(K)

∫∫

QnK

vh,ν. (61)

Note that ϕK ∈ E ; by (10) and (58) |ϕK | ≤ const and |DϕK | ≤ const/h with const

independent of h . Moreover, 1km(K)

∫∫Qn

KϕK = 1 . In addition, by the choice of ν each

control volume K ∈ T h contains a ball of radius 2ν . Hence, by (59),(60) and the definition

of π , vh|CnK≡ vh,ν |Cn

K= vnK . Therefore vh verify both equalities in (45).

By Lemma 3, (vh(·+∆x, ·)−vh(·, ·)) vanish in L1(Q) uniformly in h and ∆x ∈ B(0, ν)

as ν goes to zero. A fortiori, the same holds with the function vh replaced by vh χΩν .

Since ν → 0 as h → 0 , by the usual property of convolution regularisations (cf. e.g. [Br,

Theoreme IV.22]), ‖vh,ν − vh χΩν‖L1(Q) → 0 as h→ 0 . Moreover, it follows by the Holder

inequality, the discrete Poincare inequality (cf. Lemma 9 in the Appendix), and Proposition 1

that∫ T

0

∫

Ω\Ων

|vh| ≤(∫∫

Q

|vh|p)1/p (

T |Ω \ Ων |)1/p′

≤ const(|Ω \ Ων |

)1/p′→ 0

as h→ 0 . Therefore

‖vh,ν − vh‖L1(Q) → 0 (62)


as h→ 0 . Finally,

‖vh − vh,ν‖L1(Q) =∑

K,n

|αnK |∫∫

QnK

ϕK =

=∑

K,n

1

km(K)

∣∣∣∣∣

∫∫

QnK

vh −∫∫

QnK

vh,ν

∣∣∣∣∣

∫∫

QnK

ϕK ≤

≤ const∑

K,n

∫∫

QnK

|vh − vh,ν| = const ‖vh − vh,ν‖L1(Q) → 0

as h→ 0 . Therefore (44) holds.

It remains to prove (43). First, let us estimate vh,ν in E . It is convenient to write

Dρ(·) in the spherical coordinates; indeed, if |x| = r and x/|x| = eθ ∈ S(0, 1) , we

have Dρ(x) = const eθχB(0,1) a.e. on Rd . Let S+(0, r) = x ∈ S(0, r) | x1 > 0 and

B+(0, r) = x ∈ B(0, r) | x1 > 0 . Denote vh χΩν by f . Separating the two hemispheres

S+(0, 1) and S(0, 1) \ S+(0, 1) , we find

|Dvh,ν(t, x)| = 1

νd+1

∣∣∣∣∫Dρ(x− y

ν

)f(t, y) dy

∣∣∣∣ =

=1

νd+1

∣∣∣∣∫

B(0,ν)

Dρ(σν

)f(t, x− σ) dσ

∣∣∣∣ =

=const

νd+1

∣∣∣∣∫

S(0,1)

eθ

∫ ν

0

f(t, x− reθ)rd−1 drdθ

∣∣∣∣ =

=const

νd+1

∣∣∣∣∫

S+(0,1)

∫ ν

0

|f(t, x+ reθ)− f(t, x− reθ)|rd−1 drdθ

∣∣∣∣ ≤

≤ const

νd+1

∫∫

B+(0,ν)

∫ ν

0

|f(t, x+ σ)− f(t, x− σ)| dσ.

Therefore by the Holder inequality one has

∫∫

Q

|Dvh,ν|p ≤ const

νp(d+1)

∫∫

Q

∫

B+(0,ν)

|f(t, x+ σ)− f(t, x− σ)|p(const νd)p/p′ dσdxdt ≤

≤ const ν−d−p

∫

B+(0,ν)

∫∫

Q

|vh(t, x+ σ)− vh(t, x)|p dxdtdσ ≤

≤ const ν−p sup|σ|≤ν

∫∫

Q

|vh(t, x+ σ)− vh(t, x)|p dxdt.

Hence by Lemma 3, (58) and Lemma 4 we finally deduce that ‖Dvh,ν‖Lp(Q) ≤ const uni-

formly in h .

Now we are able to estimate (vh − vh,ν) in E . In fact, we have

∫∫

Q

|D(vh − vh,ν)|p =∑

K,n

|αnK |p∫∫

QnK

|DϕK|p ≤const

hp

∑

K,n

km(K)|αnK |p. (63)

2.II.6. Compactness result for the parabolic term 147

Moreover,

|αnK |p =

1

(km(K))p

∣∣∣∣∣

∫∫

QnK

(vh,ν − vh)

∣∣∣∣∣

p

≤ 1

(km(K))p(km(K))p/p

′×

×∫∫

QnK

∣∣vh,ν − vh∣∣p ≤ 1

km(K)const δ(K)p

∫∫

QnK

|Dvh,ν|p,(64)

since Dvh ≡ 0 on QnK . The inequality in (64) holds, because, as we have already seen,

vh,ν|CnK

≡ vnK ≡ vh|CnK, and the Poincare inequality for (vh,ν − vh) in K therefore holds

with const independent of h, ν . Substituting (64) into (63) and using (58), we obtain the

desired estimate and prove (43). ⋄

6 The compactness result for the parabolic term

In this section we prove the uniform estimate (46) on space translates of uh in L1(Q) ,

starting from the result of Lemma 3 in Section 5. Then we identify the function u = limh→0 uh

in L1(Q) (for a subsequence) with b(v) , where v = weak− limh→0 vh in E .

Lemma 6 Let(T h, k)


and uh be defined by (34). Then (46) holds.

Proof : It is a modification of the corresponding proofs in [AL83] and Chapter 2.I.

First note that∫∫

Q

|uh(t, x+∆x)− uh(t, x)| dxdt ≤ k

∫

Ω

|u0,h(x+∆x)− u0,h(x)| dx+

+2

∫∫

Q

|b(vh(t, x+∆x))− b(vh(t, x))|.(65)

By (27), the set u0,h is compact in L1(Q) , so that

sup|∆x|≤∆

∫

Ω

|u0,h(x+∆x)− u0,h(x)| dx ≤ ω0(∆)

with ω0 : R+ 7→ R+ such that ω0(∆) → 0 as ∆ → 0 , uniformly in h . Moreover, note that

b(vh) are equiintegrable on Q . Indeed, for all set F ⊂ Q one has by (47) and Proposition 1∫∫

F

|b(vh)| ≤ infδ>0

(δ

∫∫

Q

B(vh) + C(δ)|F |)≤ inf

δ>0

(const δ + C(δ)|F |

)= ω1(|F |),

and ω1(|F |) → 0 as |F | → 0 .

Further, for M > 0 let us introduce RhM = (t, x) ∈ Q| |vh(t, x)| ≤M, |vh(t, x+∆x)| ≤

M . It follows from Proposition 1 and the discrete Poincare inequality (Lemma 9 in the

Appendix) that ‖vh‖L1(Q) ≤ const uniformly in h . Hence |Q \ RhM | → 0 as M → +∞

uniformly in h , by the Chebyshev inequality. Let ωb,M(·) be the modulus of continuity of

b(·) on [−M,M ]N . Integrating separately over Q \RhM and Rh

M in the last term in (65),

we get∫∫

Q

|uh(t, x+∆x)− uh(t, x)| ≤ ω0(|∆x|) + 2ω1(|Q \RhM |)+

+

∫∫

RhM

ωb,M(|vh(t, x+∆x)− vh(t, x)|) dxdt.

It follows by the concavity of ωb,M(·) and Lemma 3 that

sup|∆x|≤∆

∫∫

Q

|uh(t, x+∆x)− uh(t, x)| dxdt ≤ infM>0,δ>0

ω0(|∆x|) + 2ω1(sup

h|Q \Rh

M |)+

+|Q|ωb,M

( 1

|Q|

∫∫

Q

|vh(t, x+∆x)− vh(t, x)| dxdt)

= ωx(∆),

and ωx(∆) → 0 as ∆ → 0 uniformly in h . ⋄

Lemma 7 Let(T h, k)

be an admissible family of grids and (vnK)K,n be a solution of

(Ph) . Assume that vh v in E and uh → u in L1(Q) for a sequence h → 0 , where

uh are defined by (34) and vh satisfy (43),(44). Then u = b(v) .

Proof : We claim that vh v in L1(Q) and b(vh) → u in L1(Q) , and then apply

the usual monotonicity argument (cf. [BrSt73]).

Since vh v also in L1(Q) , the first claim follows from (44). Further, let us show that

uh → u in L1(Q) implies ‖uh − b(vh)‖L1(Q) → 0 as h→ 0 . We have

‖uh − b(vh)‖L1(Q) =∑

K,n

∫∫

QnK

∣∣∣∣b(vnK) +

t− kn

k(b(vnK)− b(vn−1

K ))

∣∣∣∣ dxdt =

=1

2

∑

K,n

km(K)|b(vnK)− b(vn−1K )| ≤ 2

∑

K,n

∫∫

QnK

|uh(t + k, x)− uh(t, x)| dxdt.(66)

The last inequality follows form the “geometrical” observation that

k|β − α|+ k|γ − β| ≤ 4

∫ k

0

|α+ θ(β − α)− β − θ(γ − β)| dθfor all k > 0, α, β, γ ∈ R.

(67)

Indeed, it is easily checked that for L = |β−α|+|γ−β| fixed, the minimum of the right-hand

side of (67) is attained at (β−α) = −(γ − β) and equals 14kL . Recall (34); it is sufficient

to apply (67) to α = b(vn−1K ) , β = b(vnK) , γ = b(vn+1

K ) in order to obtain (66). Note that,

by the compactness of uh in L1(Q) , (48) holds. Therefore the right-hand side of (66)

vanishes as h→ 0 .

Without loss of generality, we can assume that b(vh) → u a.e. on Q . For each ε > 0 ,

choose a set Rε ⊂ Q , with |Rε| < ε , such that v ∈ L∞(Q \ Rε) and b(vh) → u in

L∞(Q \Rε) . This is always possible, by the Chebyshev inequality and the Egorov Theorem.

2.II.7. Consistency of the finite volume approximation 149

Consequently, we have b(v + λζ) → b(v) in L∞(Q \ Rε) as λ ∈ R tends to zero, for all

ζ ∈ L∞(Q) . Then for all η ∈ L∞(Q)

∫∫

Q\Rε

u · (v − η) = limh→0

∫∫

Q\Rε

b(vh) · (vh − η) ≥

≥ limh→0

∫∫

Q\Rε

b(η) · (vh − η) =

∫∫

Q\Rε

b(η) · (v − η).(68)

The inequality in (68) is due to the monotonicity of b(·) :∫∫Q\Rε

(b(vh)−b(η)) · (vh−η) ≥ 0 .

Now it is sufficient to take η = v + λζ with λ ↑ 0 and λ ↓ 0 in order to deduce that

±∫∫

Q\Rε(u − b(v)) · ζ ≥ 0 . Since ζ ∈ L∞(Q) and ε > 0 are arbitrary, u = b(v) a.e. on

Q . ⋄

7 Consistency of the finite volume approximation

In this section we prove that, for an arbitrary function η ∈ E , the finite volume approximation

Ahη = −divAhη defined by (52),(49) is a good approximation for Aη = div ap(Dη) in E ′ .

More exactly, we have the following definition.

Definition 5 (Consistent approximations) Let (T h) be a family of admissible meshes,

parametrized by h→ 0 ,(T h, kh)

be the corresponding grids with maxsize(T h), kh ≤

h , and Dh be a family of discrete gradient operators. Let Ah : E 7→ E ′ be the operator

defined by (52),(49).

We say that Ah is the approximation of the elliptic operator A · = −div ap(D ·) corre-

sponding to(T h, kh)

and Dh . This approximation is consistent if for all η ∈ E one

has Ahη → Aη in E ′ as h→ 0 .

7.1 Properties of finite volume approximations

of the elliptic term and the consistency

Theorem 3 (Consistency) Let the family of grids and the family of discrete gradient oper-

ators be admissible in the sense of Definitions 3 and 4, respectively. Then the corresponding

approximation of −div ap(D ·) is consistent.

The main ingredient of the proof is the following result. Recall that Υκ(QnK) denotes

the union of all space-time volumes of (T h, k) that are separated from QnK by at most κ

(space or time) interfaces.

Proposition 3 Let(T h, k)

be an admissible family of grids, and Dh be an admissible

family of discrete gradient operators. Then the operators Ah : E 7→ Lp′(Q) defined by

(51),(49) have the following properties:

(i) The operators Ah are uniformly local, i.e., there exists a constant C , independent of

h , such that for all η ∈ E , for all set H ⊂ Q such that H =⋃m

i=1QniKi

, one has

‖Ahη‖p′Lp′(H)

= ‖Dηh‖pLp(H) ≤ C ‖Dη‖pLp(Υκ+1(H)),

where Υκ+1(H) =⋃m

i=1Υκ+1(QniKi) .

(ii) The operators Ah are locally Holder equi-continuous, i.e., for all R > 0 there exists a

constant C(R) , independent of h , such that

‖Ahη −Ahµ‖Lp′(Q) ≤ C(R)‖η − µ‖αE

whenever ‖η‖E ≤ R , ‖µ‖E ≤ R . Here α = 1/2 for p ≥ 2 and α = p/(p′)2 for

1 < p ≤ 2 .

Proof of Theorem 3: We have to prove that ‖ap(Dη)−Ahη‖Lp′(Q) → 0 as h→ 0 .

Let us first prove the theorem for the case of η ∈ E that is piecewise constant in t and

piecewise affine in x . Let J ⊂ Q be the set of discontinuities of Dη . Clearly, J is of finite

d -dimensional Hausdorff measure Hd(J) . Let us introduce Hh =⋃

K,n |Υκ(QnK)∩J 6=ØQ

nK .

Note that |Hh| ≤ (κ + 1)hHd(J) → 0 as h → 0 ; likewise, |Υκ+1(Hh)| → 0 as h → 0 .

Therefore by (2) and Proposition 3(i) we have

∫∫

Hh

|ap(Dη)− ap(Dηh)|p′ ≤

∫∫

Hh

|Dη|p + C

∫∫

Υκ+1(Hh)

|Dη|p → 0

as h → 0 . Besides, for all QnK such that Qn

K ∩ Hh = Ø we have Dηh ≡ Dη on QnK .

Indeed, we have Dη ≡ const on Υκ+1(QnK) . Therefore Dhη|Qn

K≡ Dη = const by (23).

Hence Dw = Dη satisfies the boundary condition in (35); the equation is also satisfied, since

div ap(Dhη) ≡ 0 on QnK and 1

km(K)

∫∫Σn

Kap(Dhη)νK = ap(Dη)

∫∂KνK = 0 .

It follows that

‖ap(Dη)−Ahη‖Lp′ (Q) =(∫∫

Hh

|ap(Dη)− ap(Dηh)|p′

)1/p′→ 0

as h→ 0 , which was our claim.

Now let us approximate an arbitrary function η in E by functions µl that are piecewise

constant in t and piecewise affine in x . More exactly, there exists a sequence µl in E such

that µl → η in E and a.e. on Q as l → ∞ , and |Dµl|p are dominated by an L1(Q)

function independent of l . We have

‖ap(Dη)−Ahη‖Lp′(Q) ≤ ‖ap(Dη)− ap(Dµl)‖Lp′ (Q)+

+‖ap(Dµl)−Ahµl‖Lp′(Q) + ‖Ahµl −Ahη‖Lp′(Q).(69)

As l → 0 , the first term in the right-hand side of (69) converges to zero by the Lebesgue

dominated convergence theorem, independently of h . The second one converges to zero as

h → 0 , for all l fixed. Finally, by Proposition 3(i) and (ii), the third one converges to zero

as l → ∞ , uniformly in h . Hence the left-hand side of (69) can be made as small as desired

for h sufficiently small. This concludes the proof. ⋄

7.2 Proof of Proposition 3

The main ingredient is the following lemma.

Lemma 8 Let(T h, k)

be an admissible family of grids, and Dh be an admissible

family of discrete gradient operators. Let η, µ ∈ E and H ⊂ Q such that H =⋃m

i=1QniKi

.

Then

(i) for all R > 0 there exists a constant C(R) , independent of h , such that

m∑

i=1

‖ap(Dhη)− ap(Dhµ)‖p′Lp′(Σ

niKi

)≤ C(R)

h‖Dη −Dµ‖minp,p′

Lp(Υκ+1(H))

whenever ‖η‖E ≤ R , ‖µ‖E ≤ R ; here Υκ+1(H) =⋃m

i=1Υκ+1(QniKi) ;

(ii) in case µ = 0 , one has

m∑

i=1

‖ap(Dhη)‖p′Lp′(Σ

niKi

)≤ C

h‖Dη‖pLp(Υκ+1(H))

with a constant C independent of ‖η‖E .

Proof of Lemma 8: We consider separately the two cases 1 2 . Note

the following inequalities, valid for all y1, y2 ∈ (Rd)N (cf. e.g. [DiDT94, Bou97]):

|ap(y1)− ap(y2)|p′ ≤ const |y1 − y2|p, 1 < p ≤ 2;

|ap(y1)− ap(y2)|p′ ≤ const |y1 − y2|p′(|y1|(p−2)p′ + |y2|(p−2)p′

), p ≥ 2.

(70)

a) 1 < p ≤ 2 . We first claim that, i = 1, . . . , m , (70) and (20) yield


niKi

)≤ const

h‖Dh(η − µ)‖p

Lp(QniKi

). (71)

Indeed, note that for arbitrary n and K , for all σ ∈ EK one has

∫ nk

(n−1)k

∫

σ

|Dh(η − µ)|p = d

dK,σ

∫ nk

(n−1)k

∫

S(K,σ)

|Dh(η − µ)|p

by (18); besides, 1/dK,σ ≤ ζ/h by (10).


Combining (71) with (22), Corollary 1 (cf. the Appendix) and (9), we obtain


niKi

)≤

≤ const

h‖Dh

⊥(η − µ)‖pLp(Υκ(Q

niKi

))≤ const

h‖Dη −Dµ‖p

Lp(Υκ+1(QniKi

)).

Summing over i from 1 to m and using (9) once again, we get (i). Note that the constant

is independent of ‖η‖E, ‖µ‖E , so that (ii) holds in this case.

b) p > 2 . As in case a), we get from (18) and (10)

m∑

i=1


niKi

)≤ const

h

m∑

i=1

‖ap(Dhη)− ap(Dhµ)‖p′Lp′(Q

niKi

).

Further, by (70) and the Holder inequality with q = p/p′ and q′ = (p− 1)/(p− 2) we get

m∑

i=1


niKi

)≤ const

h

( m∑

i=1

‖Dhη −Dhµ‖p′/pLp′(Q

niKi

)

)p′/p×

×( m∑

i=1

‖Dhη‖pLp′(Q

niKi

)+

m∑

i=1

‖Dhµ‖pLp′(Q

niKi

)

)(p−2)/(p−1)

≤

≤ const

hRp(p−2)/(p−1)

( m∑

i=1

‖Dhη −Dhµ‖p′/pLp′(Q

niKi

)

)p′/p.

As in case a), (i) follows by (22), Corollary 1 and (9):

m∑

i=1


niKi

)≤ const

hRp(p−2)/(p−1)‖Dη −Dµ‖p′Lp(Υκ+1(H)).

Moreover, in case µ = 0 we have directly

m∑

i=1


niKi

)≤ const

h

m∑

i=1

‖Dhη‖pLp(Q

niKi

)≤ const

h‖Dη‖Lp(Υκ+1(H)),

so that (ii) also holds, which ends the proof. ⋄

Proof of Proposition 3:

(i) Take a grid volume QniKi

⊂ H . We have

Ahη|QniKi

= ap(Dηh)|Qni

Ki

= ap(DηniKi),

where DηniKi

is the solution of the analogue of (35) with AniKi

= ap(Dhη)|∂Ki.

Let ηh be the time average of η in each grid volume:

ηh(t, ·)|QnK=

1

k

∫ nk

(n−1)k

η(τ, ·) dτ for all K, n.

Note that on each QnK , both ηh, ηh do not depend on t . It follows from Lemma 1 that

1

p‖Dηh‖p

Lp(QniKi

)−∫∫

ΣniKi

ηhap(Dhη)νKi≤ 1

p‖Dηh‖p

Lp(QniKi

)−∫∫

ΣniKi

ηhap(Dhη)νKi.

Summing over i from 1 to m , by the Holder inequality we obtain

‖Dηh‖pLp(H) ≤ ‖Dηh‖pLp(H) + p( m∑

i=1

‖ηh − ηh‖pLp(Σ

niKi

)

)1/p( m∑

i=1


niKi

)

)1/p′(72)

in the sense of traces. Applying the Poincare inequality and the imbedding theorem (cf.

Lemma 11 and Lemma 12 in the Appendix, respectively), we obtain


niKi

)≤ const

( 1

δ(K)δ(K)p‖Dηh −Dηh‖p

Lp(QniKi

)+

+δ(K)p−1‖Dηh −Dηh‖pLp(Q

niKi

)

)≤ const hp−1‖Dηh −Dηh‖p

Lp(QniKi

).

Note that ‖Dηh‖Lp(H) ≤ ‖Dη‖pLp(H) . Therefore

( m∑

i=1


niKi

)

)1/p≤ const h1/p

′(‖Dηh‖pLp(H) + ‖Dη‖pLp(H)

)1/p. (73)

Taking into account Lemma 8(ii) and applying twice the Young inequality ab ≤ δap+C(δ)bp′,

valid for all a, b ∈ R+, δ > 0 and the corresponding C(δ) , we get from (72),(73)

‖Dηh‖pLp(H) ≤ ‖Dη‖pLp(H)+

+const h1/p′(‖Dηh‖pLp(H) + ‖Dη‖pLp(H)

)1/ph−1/p′‖Dη‖p/p′Lp(Υκ+1(H)) ≤

≤ 1

2‖Dηh‖pLp(H) + const ‖Dη‖pLp(Υκ+1(H));

whence (i) is immediate.

(ii) First note that, by (i), we can assume

‖Dη‖Lp(Q), ‖Dηh‖Lp(Q), ‖Dµ‖Lp(Q), ‖Dµh‖Lp(Q) ≤ R (74)

for some R > 0 , uniformly in h .

Fix a grid volume QnK . Taking (ηh−µh) as a test function in (35) written for ηh , then in

(35) written for µh , subtracting the two identities and integrating in t over ((n−1)k, nk) ,

we get∫∫

QnK

(ap(Dηh)− ap(Dµ

h)) : (Dηh −Dµh) = − 1

km(K)

∫∫

ΣnK

(ap(Dhη)− ap(Dhµ)) :

:

∫∫

QnK

(ηh − µh) +

∫∫

ΣnK

(ηh − µh)(ap(Dhη)− ap(Dhµ))νK .(75)

Consider separately the two cases 1 2 . Note the following inequalities,

valid for all y1, y2 ∈ (Rd)N (e.g., cf. [DiDT94, Bou97]):

|ap(y1)− ap(y2)|p′ ≤ const (ap(y1)− ap(y2)) : (y1 − y2), 1 < p ≤ 2;

|ap(y1)− ap(y2)|p′ ≤≤ const

[(ap(y1)− ap(y2)) : (y1 − y2)

]p′/2[|y1|p + |y2|p

](2−p′)/2

, p ≥ 2.

(76)

a) 1 < p ≤ 2 . This time, let us normalize ηh, µh on QnK so that

∫∫

QnK

(ηh − µh) = 0 .

By (75) and (76), we obtain

∫∫

Q

|ap(Dηh)− ap(Dµh)|p′ ≤

≤ const(∑

K,n

‖ap(Dhη)− ap(Dhµ)‖p′Lp′(Σn

K)

)1/p′(∑

K,n

‖ηh − µh‖pLp(ΣnK)

)1/p′.

As in the proof of (i), we find out that

∑

K,n

‖ηh − µh‖pLp(ΣnK) ≤ const h1/p

′‖Dηh −Dµh‖pLp(Q). (77)

On the other hand, Lemma 8(ii) yields

∑

K∈T h

‖ap(Dηh)− ap(Dµh)‖p′Lp(Σn

K) ≤C(R)

h‖Dη −Dµ‖pLp(Q). (78)

Substituting (77),(78) into (75) and taking into account (74), we deduce that

‖ap(Dηh)− ap(Dµh)‖p′

Lp′ (Q)≤ C(R) h−1/p′‖Dη −Dµ‖p/p′Lp(Q)×

×h1/p′‖Dηh −Dµh‖Lp(Q) ≤ C(R)‖Dη −Dµ‖p/p′Lp(Q).

Thus Ah are locally Holder equi-continuous with α = p/p′2 .

b) p > 2 . Using (76),(75),(74), and applying the Holder inequality with q = 2/p′ ,

q′ = 2/(2− p′) , we infer

∫∫

Q

|ap(Dηh)− ap(Dµh)|p′ ≤

(∫∫

Q

(ap(Dηh)− ap(Dµ

h)) : (Dηh −Dµh))p′/2

×

×(|Dηh|p + |Dµh|p

)(2−p′)/2

≤ C(R)(∑

K,n

‖ap(Dhη)− ap(Dhµ)‖p′Lp′(Σn

K)

)1/p′×p′/2

×

×(∑

K,n

‖ηh − µh‖pLp′(Σn

K)

)1/p×p′/2

.

As in case a), we deduce

‖ap(Dηh)− ap(Dµh)‖p′

Lp′(Q)≤ C(R)‖Dη −Dµ‖p′/2Lp(Q).

Thus Ah are locally Holder equi-continuous with α = 1/2 . ⋄

2.II.8. Appendix 155

Appendix: auxiliary results

In this appendix, we give some useful auxiliary results. First, we prove a discrete version of

the Poincare inequality for proportional meshes.

Lemma 9 (Discrete Poincare inequality) Let T be an admissible mesh of Ω ⊂ Rd in

the sense of Definition 2, δ(Ω) = diamΩ , h = size(T ) , and ζ be defined by (10). Let

(wK)K be a set of values in RN such that wK = 0 for all K ∈ Text . Then there exists

α > 0 depending only on p , d and ζ such that

∑

K∈T h

m(K) |wK |p ≤1

αδ(Ω)p

∑

(K,L)∈Υ

1

dm(K|L) dK,L

∣∣∣∣wL − wK

dK,L

∣∣∣∣p

.

Proof : We follow the proof of the Poincare inequality for W 1,p0 (Ω) . Without loss of

generality, one can assume that Ω ⊂ [0, δ(Ω)]d . As in the proof of Lemma 3, for all x ∈ Ω

consider the intersection of the segment [x, x+∆x] with the control volumes of T , where

∆x is chosen to be (−δ(Ω), 0, . . . , 0) ∈ Rd . Let Bx,x+∆x be the corresponding broken line,

and lx,x+∆x its length; one has lx,x+∆x ≤ l(δ(Ω)) in the notation of Lemma 3. Define w

by w|K = wK . Since the boundary condition on w is zero, as in Lemma 3 we have

|w(x)|p ≤ l(δ(Ω))p−1∑

(K,L)∈Υ | [x,x+∆x]∩(K|L)6=Ø

dK,L

∣∣∣∣wK − wL

dK,L

∣∣∣∣p

.

This time, |ΩKL| ≤ δ(Ω)m(K|L) , where ΩKL = x ∈ Rd | [x, x + ∆x] ∩ (K|L) 6= Ø .

Hence∫

Ω

|w|p ≤ l(δ(Ω))p−1∑

(K,L)∈Υ

dK,L

∣∣∣∣wK − wL

dK,L

∣∣∣∣p

|ΩKL| ≤

≤ δ(Ω)l(δ(Ω))p−1d∑

(K,L)∈Υ

1

dm(K|L) dK,L

∣∣∣∣wK − wL

dK,L

∣∣∣∣p

.

Our claim now follows from Lemma 4. ⋄

Remark 7 As in Remark 6, a more careful estimate of |ΩKL| in the proof of Lemma 9 allows

to bypass (10), in case p ≤ 2 . For p = 2 , this yields the result of [YCGH, Remark 7].

Next, we prove another kind of Poincare inequality, under the form convenient for appli-

cation in finite volume schemes (cf. [EGGHH00, Lemma 6.1]).

Lemma 10 Let K be a volume of an admissible mesh of Ω ⊂ Rd in the sense of Definition 2,

and QnK = ((n− 1)k, nk) ×K be the corresponding element of the space-time grid. Then

there exists a constant C that only depends on p , d , and ζ in (11) such that for all

σ ∈ EK , for all w ∈ Lp((n− 1)k, nk;W 1,p(K)) one has

1

dkm(σ) dK,σ

∣∣∣∣wK − wσ

dK,σ

∣∣∣∣p

≤ C ‖Dw‖pLp(QnK) , (79)


where wK = 1km(K)

∫∫Qn

Kw and wσ = 1

km(σ)

∫ nk

(n−1)k

∫σw , in the sense of traces. Besides,

the same holds with wK replaced by wC = 1|Cn

K |

∫∫CnKw , where Cn

K is defined in Section 1.

Proof : Set h = δ(K) ; one has ν = ζδ(K)/(2√d) . First, note that

1

dkm(σ) dK,σ

∣∣∣∣wK − wσ

dK,σ

∣∣∣∣p

≤ C0(p, d, ζ)m(σ)

hp−1k |wK − wσ|p.

Further, it follows by the Jensen inequality that

k |wK − wσ|p ≤∫ nk

(n−1)k

|wK(t)− wσ(t)|p,

where wK(t) , wσ(t) are the averages of w(t, ·) over K and σ , respectively, for a.a.

t ∈ ((n− 1)k, nk) . Thus it is sufficient to prove that for all w ∈ W 1,p(K) one has

|wK − wσ|p ≤ consthp

m(K)‖Dw‖pLp(K) (80)

(we abusively keep the same notation, dropping the dependence of w on t ).

Furthermore, without loss of generality we can assume that σ is parallel to the hyperplane

x1 = 0 of Rd . Let C = C1 × Cd−1 be a d -dimensional cube with edge 2ν and sides

parallel or perpendicular to σ , contained in K , and such that dist (C, σ) ≥ ν (see fig.2).

Let m(C) denote the d -dimensional measure of C ; set wC = 1m(C)

∫Cw . It is known that

|wK−wC |p ≤ const hp

m(K)‖Dw‖Lp(K) (cf. (84) in Lemma 11 below). Therefore it is sufficient

to prove (80) with wK replaced by wC ; this will also prove the last statement of the lemma.

We denote a point of σ by s and a point of C by x = (ρ, l) , ρ ∈ C1 , l ∈ Cd−1 . By

the standard density argument, we can assume that w ∈ C1(K) . By the Newton-Leibnitz

formula and the Holder inequality one has

|wC − wσ|p ≤1

(m(C)m(σ))p

(∫

C

∫

σ

∫ |x−s|

0

∣∣∣∣Dw(s+ rx− s

|x− s|)∣∣∣∣ drdsdx

)p≤ hp−1

m(C)m(σ)×

×∫

C

∫

σ

∫ |x−s|

0

∣∣∣∣Dw(s+ rx− s

|x− s|)∣∣∣∣p

drdsdx ≤ hp−1

m(σ)

const

hd×

×∫ ∫ ∫

(ρ,l,s,r)∈C1×Cd−1×σ×[0,h]

∣∣∣∣Dw(s+ r

x− s

|x− s|)∣∣∣∣p

drdsdldρ ≤

≤ hp−1

m(σ)

const

hd

(I1 + I2

),

(81)

where I1, I2 are the integrals of |Dw(s+ r x−s|x−s|)|p over M ∩r ∈ [0, ν/2] and M ∩r ∈

[ν/2, h] , respectively, and M = C1 × Cd−1 × σ × [0, h] . In (81) we have extended Dw by

zero outside K , so that all integrals make sense.

Introduce the change of variables (ρ, l, s, r) ↔ (ρ, s, y) , where y = s+ r x−s|x−s| (cf.

fig.2). Clearly, for all (ρ, s) ∈ C1 × σ there is a one-to-one correspondence between (l, r) ∈Cd−1 × [0, h] and y in some subset of K . Moreover, the Jacobian det ‖D(l,r)

Dy‖ of the

K

σ

C

x ∈ Cρ ∈ C1

l ∈ Cd−1

y ∈ K

s ∈ σr = |y − s|

x1 ∈ R

(x2, . . . , xd) ∈ Rd−1

K

σ

Cyσy

ν/2 ≥ν/2

2ν

Figure 2: The cube C and the change of variables Figure 3: The set σy

transformation (l, r) ↔ y is estimated by const(h/|y1|

)d−1

, where const is independent

of (ρ, l, s, r) ∈M . Thus we have

I2 ≤∫

y∈K

(|Dw(y)|p

∫∫

s∈σ,ρ∈C1

dsdρ

)const dy ≤ hm(σ)‖Dw‖pLp(K). (82)

On the other hand, one has

I1 ≤∫

y∈K

(|Dw(y)|p

∫

s∈σ | ∃x∈C,−→x−s ‖−→y−s

ds

∫

ρ∈C1

dρ

)const

( h

|y1|)d−1

dy

Note that, when |y1| < ν/2 , the (d−1) -dimensional measure of the set σy = s ∈ σ | ∃x ∈C, −−−→x− s ‖−−−→y − s is estimated by const |y1|d−1 , where the constant is absolute (cf. fig.3).

Hence

I2 ≤ const hd−1h

∫

y∈K

|Dw(y)|p 1

|y1|d−1|y1|d−1 dy ≤ const hd ‖Dw‖pLp(K). (83)

Substituting (82),(83) in (81), and taking into account that m(σ) ≤ hd−1 , we finally deduce

(80). This ends the proof. ⋄

Corollary 1 Let (T h, k) be a family of admissible space-time grids in the sense of Defini-

tion 3, and let Dh⊥ be the corresponding operator defined by (19). Let η ∈ E . Then there

exists a constant C which only depends on p , d , and M , ζ such that

∥∥Dh⊥η∥∥pLp(Qn

K)≤ C ‖Dη‖pLp(Υ1(Qn

K)) .

Proof : Let ηnσ = 1km(σ)

∫ nk

(n−1)k

∫ση . By definition, we have

∥∥Dh⊥η∥∥pLp(Qn

K)=

∑

σ∈EKσ=K|L

1

dkm(σ) dK,σ

∣∣∣∣ηnL − ηnKdK,L

∣∣∣∣p

≤ C∑

σ∈EKσ=K|L

1

dkm(σ) dK,σ

( |ηnσ − ηnK |p(dK,σ)

p +|ηnσ − ηnL|p

dK,σ (dL,σ)p−1

)

≤ C∑

σ∈EK

1

dkm(σ) dK,σ

∣∣∣∣ηnσ − ηnKdK,σ

∣∣∣∣p

+ C∑

σ∈EKσ=K|L

1

dkm(σ) dL,σ

∣∣∣∣ηnσ − ηnLdL,σ

∣∣∣∣p

≤ C ‖Dη‖pLp(Υ1(QnK)) ,

by Lemma 10. ⋄

Finally, we need the following versions of the Poincare inequality the trace imbedding

theorem for the spaces W 1,p .

Lemma 11 Let K ⊂ Rd be convex, bounded of diameter δ(K) and contain a cube C of

edge ζδ(K) > 0 ; let 1 ≤ p < ∞ . Let w ∈ W 1,p(K) , wK = 1|K|

∫Kw . Then there exists

a constant C = C(p, d, ζ) , independent of w , such that

‖w − wK‖Lp(K) ≤ C δ(K) ‖Dw‖Lp(K) .

In addition, one has

|wK − wC|p ≤ Cδ(K)p

|K| ‖Dw‖Lp(K) , (84)

where wC = 1|C|

∫Cw , with a constant C = C(p, d, ζ) , independent of w .

Proof : The proof follows the lines of the proofs of [EgKo, Theorems 59,60], with p = 2

replaced by p ∈ (1,∞) and the Cauchy inequality replaced by general Holder inequality. ⋄

Lemma 12 Let K ⊂ Rd be convex, bounded of diameter δ(K) and contain a ball of radius

ζδ(K) > 0 ; let 1 ≤ p <∞ . Then there exists a constant C which only depends on p and

ζ such that

‖w‖pLp(∂K) ≤ C

(1

δ(K)‖w‖pLp(K) + δ(K)p−1 ‖Dw‖pLp(K)

),

for all w ∈ W 1,p(K) .


Proof : We cannot refer to [EgKo, Theorem 76], where the boundary of K is supposed

to be Lipschitz. Nevertheless, it is sufficient to introduce coordinates of the spherical type in

a neighbourhood of ∂K of thickness of order δ(K) . More exactly, let xK be the center

of the ball of radius ζδ(K) inside K . Introduce the family of (d−1) -dimensional surfaces

Sτ ⊂ K homothetic to S0 = ∂K with respect to xK . Parametrize it by τ the maximal

distance between the points of Sτ and S0 lying on the ray emanating from xK that have

the longest intersection with K . Note that the surfaces Sτ are well defined at least for

τ ∈ [0, ζδ(K)] . Note also that the distance between the points of Sτ1 and Sτ2 lying on a

same ray emanating from xK does not exceed |τ1− τ2| . The proof of the lemma goes on as

in [EgKo, Theorem 76], upon replacing the original family Sτ by the one constructed above.

⋄

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