revisited and saturated solutions Scalar conservation laws: Initial and boundary value problems C.R. Acad. Sci. Paris, Ser.I

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Partial differential equations

Scalar conservation laws: Initial and boundary value problems revisited and saturated solutions ^✩

Lois de conservation scalaires : retour sur les problèmes aux limites et solutions saturées

Pierre-Louis Lions

^a

^,

^b

, Panagiotis Souganidis

^c

aCollègedeFrance,11,placeMarcelin-Berthelot,75005Paris,France

bCEREMADE,UniversitéParis-Dauphine,placeduMaréchal-de-Lattre-de-Tassigny,75016Paris,France cDepartmentofMathematics,UniversityofChicago,5734S.UniversityAve.,Chicago,IL60637,USA

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received25September2018 Accepted27September2018 Availableonline6November2018 PresentedbyPierre-LouisLions

Werevisittheclassicaltheoryofmultidimensionalscalarconservationlaws.Wereformu- latethenotionoftheclassicalKruzkoventropysolutionsandstudysomenewproperties as well as the well-posedness of the initial value problem with inhomogeneous fluxes and general initial data. We also consider Dirichlet boundary value problems. We put forwardanewand transparentdefinition forsolutionsandgiveasimpleproof fortheir well-posednessindomainswith smoothboundaries.Finally, weintroducethenotion of saturatedsolutionsandshowthatitiswell-posed.

©^{2018PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.Thisisan} openaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r é s um é

Nous revenons sur la théorie classique des lois de conservation scalaires multidimen- sionnelles. Nous introduisons unenotion nouvelle de sous- etde sur-solutions, qui est équivalenteàlanotionclassiquedesolutionsentropiquesàlaKruzkov.Nousutilisonsen- suitecettenotionpourétablir despropriétésnouvellesdecessolutions.Nousproposons égalementuneformulationnouvelleetclairedessolutionspourlesproblèmesauxlimites etnousdonnonsunepreuvesimpleducaractèrebienposédecesproblèmes.Enfin,nous introduisonslanotiondesolutionssaturéesetmontronsquedetelsproblèmessontbien posés.

©^{2018PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.Thisisan} openaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

✩ PartiallysupportedbytheNationalScienceFoundationgrantsDMS-1600129andtheOfficeofNavalResearch grantN000141712095.

E-mailaddresses:[email protected](P.-L. Lions),[email protected](P. Souganidis).

https://doi.org/10.1016/j.crma.2018.09.010

1631-073X/©^{2018PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense} (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Versionfrançaiseabrégée

Nous considérons dans cette Note les lois de conservation scalaires multidimensionnelles du type ut

+

^divxf

(

u

) =

⁰ dans U

× (

0

,

T

),

u

(

x

,

0

) =

^u0 dans U,où T

>

0 est fixé, d

≥

^{1, f}

= (

f₁

, . . . ,

f_d

)

, où les fonctions f_i (de R ^dans R^{) sont} supposées être de classe C¹ (pour simplifier la présentation) etla condition initiale u0 estsupposée appartenir à

(

L¹

∩

L^∞

)(

U

)

(ànouveaupoursimplifierlaprésentation).L’inconnueu

(

x

,

t

)

prendsesvaleursdansR^{etserasupposéeappartenir} àC

( [

⁰

,

T

];

^L1

(

U

)) ∩

^L∞

(

U

× (

0

,

T

))

.Quelquesinformationsbibliographiquessontdonnéesdanslaversionenanglaisquisuit.

Nous introduisons la notion de sous-solution (respectivement sur-solution) entropique en imposant que, pour tout k

∈

R

,

u

∨

^k

=

^max

(

u

,

k

)

(respectivementu

∧

^k

=

^min

(

u

,

k

)

)vérifieausensdesdistributions

(

u

∨

k

)

t

+

divxf

(

u

∨

k

) ≤

0 (resp.

(

u

∧

k

)

t

+

divxf

(

u

∧

k

) ≥

0) dans

R

^d

× (

0

,

T

)

.

Enfin, uestunesolutionentropiquesielleestàlafoissous-solutionetsur-solutionentropique.

OnvérifiealorsaisémentquecettenotionestéquivalenteàlanotiondesolutionentropiqueàlaKruzkov.Etonobserve quelaméthodededoublementdevariablesdeKruzkovpermetd’établirlerésultatsuivant.

Théorème0.1.Siu

,

v

∈

^C

([

⁰

,

T

];

^L1

(R

^d

)) ∩

^L∞

(R

^d

× (

0

,

T

))

sontdeuxsous-solutionsentropiques,alorsu

∨

^{v l’estégalement.}

Grâceàlaconservationde

R^du

(

x

,

t

)

dx,cethéorèmeimpliqueàl’évidencetouslesrésultatsclassiquesdelathéoriede Kruzkov(unicité,principedecomparaison,contractiondansL¹).

DanslecasoùU estborneé,onconsidèreleproblèmeauxlimites

ut

+

divxf

(

u

) =

0 dans U

× (

0

,

T

),

u

(·,

0

) =

u0 dans U

,

u

=

0 dans

∂

U

× (

0

,

T

),

eton introduitlanotionsuivante desous- etsur-solutionentropique :pour u

∈

^C

( [

⁰

,

T

];

^L1

(

U

)) ∩

^L∞

(

U

× (

0

,

T

))

,on dit queuestsous-solution(resp.,sur-solution)entropiquesiona,pourtoutk

∈

R^,

(

u

∨

^k

)

t

+

^divxf

(

u

∨

^k

) ≤

^{0 dansU}

× (

0

,

T

),

(

resp.,

(

u

∧

^k

)

_t

+

^divxf

(

u

∧

^k

) ≥

^{0 dans U}

× (

0

,

T

), )

etpourtoutk

≥

^{0 (resp.k}

≤

⁰⁾

(

u

∨

k

)

t

+

divxf

(

u

∨

k

) −

f1

(

k

)δ

0

(

x1

) ≤

0 dans U

× (

0

,

T

), (

resp.

(

u

∧

^k

)

t

+

^divxf

(

u

∧

^k

) −

^f1

(

k

)δ

0

(

x1

) ≥

^{0 dans U}

× (

0

,

T

))

oùl’inégalitésur U signifieque,pourtoutk

≥

^{0 et}

ϕ ∈

^C1

(

R^d

), ϕ _≥

0,àsupportcompactdansU,ona d

dt

U

(

u

∨

^k

) ϕ

dx

−

U

f

(

u

∨

^k

) ·

^D

ϕ

dx

+

∂U

f₁

(

k

) ϕ (

0

,

x

)

dx

≤

⁰

(resp.,pourtoutk

≤

^{0 et}

ϕ ∈

^C1

(

R^d

), ϕ _≥

0,àsupportcompactdansU,ona d

dt

U

(

u

∧

^k

) ϕ

dx

−

U

f

(

u

∧

^k

) ·

^D

ϕ

dx

+

∂U

f₁

(

k

) ϕ (

0

,

x

)

dx

≥

⁰

.)

Enfin, uestunesolutionentropiquesiuestàlafoissous- etsur-solutionentropique.

OndémontrealorsleThéroème 2.

Théorème0.2.i)Siu (resp.,v)

∈

^C

( [

⁰

,

T

];

^L1

(

U

) ∩

^L∞

(

U

× (

0

,

T

))

estsous-solution(resp.sur-solution)entropique,alorsona,pour toutt

∈ [

⁰

,

T

]

U

(

u

(

x

,

t

) −

^v

(

x

,

t

))

₊dx

≤

U

(

u

(

x

,

0

) −

^v

(

x

,

0

))

₊dx

ii)Siu0

∈ (

L¹

∩

^L∞

)(R

^d₋

)

,ilexisteuneuniquesolutionentropiqueu

∈

^C

([

⁰

,

T

];

^L1

(

U

) ∩

^L∞

(

U

× (

0

,

T

))

vérifiantu

(.,

0

) =

^u0p.p.

surU etu

=

^0sur

∂

U

× (

0

,

T

)

.

Signalonsqueladémonstrationdel’existenceprouvelaconvergencedel’approximationparviscositéévanescente.Pour simplicitenousprenonsU

=

R^d₋^.

Enfin,nousintroduisonsunenotionnouvelle,quenousappelons« solutionsaturée » danslecasoùU

=

R^d₋^{.Parexemple :} u

∈

^L∞

(

0

,

T

;

^L1x1

(

L^∞_x

)) ∩

^C

([

⁰

,

T

];

^L1x1

(

L¹_loc,x

)) ∩

^L∞

(R

^d₋

× (

0

,

T

))

estsolutionsaturéesielleestsous-solution entropiquesur R^d₋^etsur-solutionentropiquesurR^d₋^pourtoutk

∈

R^,i.e.

(

u

∨

k

)

t

+

f

(

u

∨

k

) +

f1

(

k

)δ

0

(

x1

) ≥

0 sur

R

^d₋

× (

0

,

T

)

pourtoutk

∈

R.Etondémontrelethéorèmesuivant.

Théorème0.3.Siu0

∈

^L1x1

(

L^∞_x

) ∩

^L∞x etsilefluxf1vérifielaconditionsuivante

∃ ζ

n

− →

n

+∞, ∀ζ ≥ ζ

n

,

f₁

(ζ

n

) ≥

^f1

(ζ ),

alors il existe une unique solution saturée. De plus, u

≥

^{v p.p. dans} R^d₋

× (

0

,

T

)

pour toute sous-solution entropique u

∈

L^∞

(

0

,

T

;

^L1x1

(

L^∞_x

)) ∩

^C

( [

⁰

,

T

];

^L1x1

(

L¹_loc,x

)) ∩

^L∞

(

R^d₋

× (

0

,

T

))

surR^d₋

× (

0

,

T

)

vérifiantv

(

x

,

0

) ≤

^u0

(

x

)

p.p.dansR^d₋^.

1. Introduction

We introducein section2 asimple formulationofsub- and super-solutions formulti-dimensionalscalarconservation laws.Thisnewformulation, whichisequivalent totheclassicalKruzkov-entropy solutions,istechnicallyveryflexibleand allows usto studysome new properties of the solutions. We also study the existence and uniqueness of solutions for generalinhomogeneousfluxesandinitialdata inL¹

+

^{L∞.In section3,we considerinitial boundaryvalue problemsand} putforwarda newdefinition,which istransparent,avoidstheneed tousespecialentropiesandtraceson theboundary, andyieldsarathereasy proofofthecomparisonprinciple.Finally,insection4we introducethenewnotionofsaturated solutions,whicharetheanaloguesofstate-constraintsolutionsforHamilton–Jacobiequations,andprovethattheyarewell posedandmaximal.

Due to page limitations,we only state withsketches of proofs some of the resultswithout any attemptto optimize assumptions.Detailscanbefoundinaforthcomingpaperoftheauthors [6] aswellasthelecturesofthefirstauthorinthe fallof2017atthe“CollègedeFrance”.SomeoftheresultswereannouncedinalectureofthefirstauthorattheUniversity ofChicagoinJanuary2017.

2. Theinitialvalueproblem

Weintroduce asimple formulationforsolutionstothe initialvalue problemforscalarconservationlaws,whichis,of course,equivalenttotheclassicalKruzkoventropysolution.

ForafixedT

>

0,weconsidertheequation

ut

+

^divxf

(

u

,

x

) =

^{0 in}

R

^d

× (

0

,

T

),

(1)

with

f

= (

f₁

, . . . ,

f_d

) ∈ (

W_loc^1,1_,x

(

C_u

))

^d

;

⁽²⁾

wesaythat f

:

R

×

R^d

→

R^d

∈ (

W_loc^1,1_,x

(

Cu

))

^d if,forall R

>

0, sup

|u|≤R,1≤i,j≤d

| ∂

fi

∂

xj

| ∈

^L1_loc

(R

^d

)

.

Thenewformulation.Inwhatfollows,a

∧

^b

=

^min

(

a

,

b

)

,a

∨

^b

=

^max

(

a

,

b

)

andsign₊z

=

^1ifz

≥

^{0and 0otherwise.More-} over,1 denotesthecharacteristicfunctionoftheset A.Finally,we writeD

(

U

)

tosignifythat anexpressionholdsinthe senseofdistributionsinU.

Definition2.1.Afunctionu

∈

^L∞_loc

(

R^d

× (

0

,

T

))

isalocalsub- (resp.super-)solutionto(1) if,foreveryk

∈

R^,

(

u

∨

^k

)

t

+

^divxf

(

u

∨

^k

,

x

) −

¹_{u<k}div_xf

(

k

,

x

) ≤

^{0 in} D

(R

^d

× (

0

,

T

)),

(3)

(

resp.

(

u

∧

^k

)

t

+

^divxf

(

u

∧

^k

,

x

) −

¹_{u>k}div_xf

(

k

,

x

) ≥

^{0 in} D

(R

^d

× (

0

,

T

)).

(4) Weremarkthatiteasytocheckthat(3) (resp.(4))alsoholdsif1_{u<k}(resp.1_{u>k})isreplacedby 1_{u≤k}(resp.1_{u≥k}).

Moreover,itisstraightforwardtoseethatDefinition2.1isinfactequivalenttotheclassicalKruzkov([5])entropysub- and super-solution.

Thefollowingresult,whichisanimmediateconsequenceofDefinition2.1,isneverthelesssurprisinglynewinthetheory ofscalarconservationlaws.

Proposition2.2.Themaximum(resp.minimum)oftwolocalsub- (resp.super-)solutionsto(1)inL^∞_loc

(

R^d

× (

0

,

T

))

isalocalsub- (resp.super-)solution.

Proof. Weonlysketchtheproofofthesub-solutionproperty,sincetheotherclaimfollowssimilarly.Theargumentisbased ontheclassical“doublingthevariables”techniqueintroducedinthetheoryofscalarconservationlawsin[5].Forsimplicity, weonlydiscussthecaseofhomogeneousfluxes.

Fix k

∈

R^and,for

(

y

,

s

), (

x

,

t

) ∈

R^d

× (

0

,

T

)

,usethedefinitionofthelocalsub-solutionforu withconstant v

(

y

,

s

) ∨

^k andforv withconstantu

(

x

,

t

) ∨

^{k.Itfollowsthat,a.e.in}

(

y

,

s

)

and

(

x

,

t

)

,

(

u

∨ (

v

(

y

,

s

) ∨

^k

))

t

+

^divxf

(

u

∨ (

v

(

y

,

s

) ∨

^k

)) ≤

^{0 in} D

( R

^d

× (

0

,

T

)),

and

(

v

∨ (

u

(

x

,

t

) ∨

^k

))

s

+

^divyf

(

v

∨ (

u

(

x

,

t

) ∨

^k

)) ≤

^{0 in} D

(R

^d

× (

0

,

T

)).

Employing

ρ

_η

(

t

−

^s

)φ

_θ

(

x

−

^y

)ψ(

^x

+

^y 2

,

^t

+

^s

2

)

as the test functions above, where

ρ

_η and

φ

_θ are approximations of the identity,afterintegratinginx

,

y

,

t

,

sandpassingtothelimits

η , θ →

^0,wefind

T 0

R^d

[(

^u

∨

^v

∨

^k

)

₊

ψ

t

−

^f

(

u

∨

^v

∨

^k

)) ·

^Dx

ψ ],

^dxdt

≤

⁰

,

and,hence,theclaim. 2

WestatenextwithoutproofanimmediateconsequenceofProposition2.2.

Corollary2.3.Ifu

,

v

∈

^L∞_loc

(R

^d

× (

0

,

T

))

arelocalsub- (resp.super-)solutionsto(1),thenu

∨

^{v (resp.u}

∧

^v)isalocalsub- (resp.

super-)solutionto(1)inthesenseofdistributionsinR^d

× (

0

,

T

)

.

Aregularization.Wediscussaregularizationofentropysub- andsuper-solutionsthatresemblestheso-calledsup- andinf- convolutionsofviscositysolutions.

Noticethat,ifu

∈

^Ct

(

L¹_x

) ∩

^L∞x,t isanentropysub- (resp.super-)solutiontothehomogeneousproblem

u_t

+

^divxf

(

u

) =

^{0 in}

R

^d

× (

0

,

T

),

(5)

then,foreachz

∈

^Rd,u

(· +

^z

, ·)

^isasub- (resp.super-)solutionto(5).

For

>

0,let

u

(

x

,

t

) :=

^esssup_|z|≤u

(

x

+

^z

,

t

)

and u

(

x

,

t

) :=

^essinf|z|≤u

(

x

+

^z

,

t

).

(6) Itisnotclearifandinwhatsenseuεandu convergetouinthegeneralitywearediscussinghere.Theyare,however, moreregularthanuandpreservethesub- andsuper-solutionproperties.

Thefollowingholds.

Proposition2.4.Letu

∈

^L∞x,tbeasub- (resp.super-)solutionto(5).Thenu(resp.u)

∈

^L∞x,tisasub- (resp.super-)solutionto(5).

Moreover,forallt

∈ (

0

,

T

)

,u

(·,

^t

),

u

(·,

^t

) ∈

^BVx.

Proof. Wesketch thesub-solutionpropertyandrefer to[6] forthefulldetails.Also, since isfixed, wewriteu inplace ofu.

Sinceu

(

x

,

t

) ∨

^k

=

^esssup_|z|≤

(

u

(

x

+

^z

,

t

) ∨

^k

)

and,foreach z

∈

R^d,u

(· +

^z

, ·) ∨

^kisasub-solutionto(5),theclaimfollows ifweshowthatuisadistributionalsub-solutionto(5).Fixt0

∈ [

⁰

,

T

)

andconsidertheinitialvalueproblem

v_t

+

^divxf

(

v

) =

^{0 in}

R

^d

× (

t₀

,

T

)

and v

(·,

^t0

) =

^u

(·,

^t0

).

(7) Recall that, for each z

∈

R^{d, u}

(·,

^t0

) ≥

^u

(· +

^z

,

t₀

)

a.e. and u

(· +

^z

, ·)

^{is a} sub-solution to (5). It then follows from the comparisonprinciplethat v

≥

^u

( · +

^z

, · )

inR^d

× (

t0

,

T

)

,and,hence,

v

≥

^uin

R

^d

× (

t₀

,

T

).

Fortheinequalityabovetobetrue,itisnecessarytouseamoreinvolvedpropertyoftheesssup,whichisdiscussedin[6].

Note that we donot know ifu

(·,

^t0

) ∈

^L1

(R

^d

)

. Instead,we have u

(·,

^t0

) ∈

^L∞

(R

^d

)

,andit is, hence,necessaryto have comparisoninL^∞,asitisshowninthenextsection.

Nextfixanon-negative

φ ∈

^C∞c

(

R^d

× (

0

,

T

))

.Since v isanentropysolution,forh

>

0,wefindthat

R^d

v

(

x

,

t0

+

h

) −

v

(

x

,

t0

)

h

φ (

x

,

t0

)

dx

=

¹ h

t

₀+^h t0

R^d

f

(

v

(

x

,

s

)) ·

D

φ (

x

,

s

)

dxds

,

and,sincev

∈

Ct

(

L¹_x,loc

)

,

lim

h→0

1 h

t

0+h t0

R^d

f

(

v

(

x

,

s

)) ·

^D

φ (

x

,

s

)

dxds

=

d R

f

(

v

(

x

,

t₀

)) ·

^D

φ (

x

,

t₀

)

dx

.

Itfollowsthat

R^d

u

(

x

,

t₀

+

^h

) −

^u

(

x

,

t₀

)

h

φ (

x

,

t₀

)

dx

≤

¹ h

t

₀+^h t0

R^d

f

(

v

(

x

,

s

)) ·

^D

φ (

x

,

s

)

dxds

,

andthus,foreacht₀

∈ (

0

,

T

)

,

lim sup

h→0⁺

R^d

u

(

x

,

t₀

+

^h

) −

^u

(

x

,

t₀

)

h

φ (

x

,

t₀

)

dx

≤

R^d

f

(

u

(

x

,

t₀

)) ·

^D

φ (

x

,

t₀

)

dx

,

and,hence,theclaim. 2

Remark2.5.A statement similar to Proposition 2.2 and thesub- (resp. super-) solution property ofProposition 2.4, but understrongerassumptions,appearedlaterinSilvestre [11].

Thewell-posedness.The mostgeneralcomparisonresultwe knowfor(1) isforu0

∈ (

L¹

+

^L∞

)(

R^d

)

and,hence,solutions inCt

((

L¹

+

^L∞

)

x

)

.Todealwithsuchgenerality,itisnecessarytomakeassumptionsonthemodulusofcontinuityoffinu.

Forexample,forcomparisonwithsolutionsinC_t

((

L¹

+

^c

))

,theconditionisaboutthemodulusofcontinuityoffatc.The detailsappearin [6].

Here we discuss solutions in C

( [

⁰

,

T

];

^L1

(

R^d

)) ∩

^L∞x,t. To formulate the conditions on f, we need to introduce some additionalnotation.For

δ >

0 andC

>

0,let

M₂^δ

:=

⎧ ⎨

⎩

sup_|z|≤δ

|

^f

(

z

,

x

) −

^f

(

0

,

x

)|

|

^z

|

^{θ if d}

≥

^{2 with}

θ =:

^d

−

¹ d

,

sup_|z|≤δ

|

^f

(

z

,

x

) −

^f

(

0

,

x

) |

^{if d}

=

¹

,

and M_1,C

:=

^sup

|z|≤C

|

^f

(

z

,

x

) −

^f

(

0

,

z

) | ,

andrecallthat,foranyp

∈ [

¹

, ∞)

^, L_unif^p

= {

^f

∈

^Lp_loc

:

^sup

n∈Z^d

^f

|

L^p(n+^B1)

< ∞} .

Weassumethat

divxf

(

0

,

x

) ∈

^L1

( R

^d

),

(8)

and,forallfor

δ >

0 andC

>

0,ifd

≥

^2,

M₂^δ

∈

L^d_unifandM1,C

∈

L^d_unif

,

(9)

or,ifd

=

^1,

⎧ ⎨

⎩

M_1,C

∈

^L1_unif^and

(

M_1,C

(

n

+ ·))

n∈Zis uniformly integrable in

[−

¹

, +

¹

],

M^δ₂

∈

L¹_unifandM^δ₂

→

0 inL¹_unifas

δ →

0

.

(10)

Weremarkthat“M_1,C

∈

^L1_unif^and

(

M_1,C

(

n

+ ·))

n∈Zis uniformly integrable in

[−

¹

, +

¹

]

^{”isequivalentto} M1,C

∈

^closure_L¹

unif

(

L_unif^p

)

for somep

>

1

.

Finally,wealsoassume,forsomec

>

0,

div_xf

(

z

,

x

)

sign

(

z

) ≥ −

^c

(

1

+ |

^z

|)

^{for allx}

∈ R

^dandz

∈ R

^{. (11)}

Weremarkthat(8),(9) or(10),and(11) areconsiderablyweakerthan

D²_x,zf

∈

^L∞

( [−

^M

,

M

] × R

^d

)

andDxdivxf

∈

^L1x

(

C

( [−

^M

,

M

] )

for eachM

>

0 (12) whichwereusuallyassumedintheexistingliterature.

Thenewcomparisonandexistenceresultisstatednext.

Theorem2.6.Assume(2).Ifu1

,

u2

∈

^Ct

(

L¹_x

) ∩

^L∞x,taresolutionsto(1),then,forallt

,

s

≥

^0suchthatt

≥

^s,

^u1

(·,

^t

) −

^u2

(·,

^t

)

1

≤

^u1

(·,

^s

) −

^u2

(·,

^s

)

1

.

(13) Assume,inaddition,(8)and(11).Foreachu₀

∈ (

L¹

∩

^L∞

)(R

^d

)

,(1)hasasolutionu

∈

^Ct

(

L¹_x

) ∩

^L∞x,t.

The L¹-contractionfollowsfromtheassumedcontinuityintimeandthefactthatCorollary2.3yields d

dt

(

u

∨

v

)(

x

,

t

)

dx

≤

0 and

−

^d dt

(

u

∧

v

)(

x

,

t

)

dx

≤

0

,

and,hence, d dt

|

^u

−

^v

|(

^x

,

t

)

dx

≤

⁰

.

Fortheexistence,weusetheapproximateviscousinitialvalueproblem

u_,t

+

^divxf

(

u

,

x

) =

u in

R

^d

× (

0

,

T

)

and u

( · ,

0

) =

^u0

,

(14) andemploy(8) and(11) toobtainindependentof

∈ (

0

,

1

)

boundson

^u

( · ,

t

)

1 and

^u

( · ,

t

)

∞.

Then, insteadof establishinga boundon

^Du

( · ,

t

)

1 thatrequires ahypothesis like(12),we usea newstrategythat doesnotrelyonsuchastrongconditionontheflux,but,instead,isbasedonanewstabilityresult,whichwestatenext.

Theorem2.7.Let

(

un

)

n∈N

⊂

^Ct

(

L¹_x

) ∩

^L1x,tbeafamilyofsolutionsto(1)withfluxesfnsatisfying(8),(9)ifd

≥

^2or(10)ifd

=

^1,and (11)withboundsuniforminn andinitialdata

(

u0,n

)

n∈NuniformlyboundedinL¹

∩

^{L∞andsuchthatu}0,n

→

^u0inL¹

∩

^L∞.Then thereexistsasolutionu

∈

^Ct

(

L¹_x

) ∩

^L1x,tto(1)withinitialdatumu₀suchthat,asn

→ ∞

^,un

→

^{u inC}t

(

L¹_x

) ∩

^L1x,t.

The proofofTheorem2.7isbasedonshowingthat thereexistmoduli

ω

δ

(

n

,

m

)

and

α (δ)

such that,asn

,

m

→ ∞

^and

δ →

^0,

ω

δ

(

n

,

m

) →

^{0 and}

α (δ) →

^{0 and}

d dt

R^d

R^d

((

u_n

∨

^um

)(

x

,

t

) −

^um

(

y

,

t

) ∧

^un

(

y

,

t

))

₊

ρ

_δ

(

x

−

^y

)

dxdy

≤ ω

_δ

(

n

,

m

) + α (δ),

(15)

where

ρ

δ isastandardmollifier.Wereferto [6] forthedetails.

3. Theinitialboundaryvalueproblem

ForT

>

0 fixedweconsidertheinitialboundaryvalueproblem

u_t

+

^divf

(

u

,

x

) =

^{0 in U}

× (

0

,

T

),

u

=

^{h on}

,

u

(·,

⁰

) =

^u0onU

,

(16) where U

⊂

R^{d isanopen setwithsmooth, forexample C1-boundary}

∂

U andexternalnormalvector

ν

,

:= ∂

U

× [

⁰

,

T

]

^, andh

: →

R^.

We continuewithaby nomeanscomprehensivereview oftheexistingliterature forsolutions to(16). Herewe tryto mentionsomeofthemostimportantreferences.Werecognize,however,thatwemayhavemissedsome.

The studyofboundaryvalueproblemsforscalarconservationlawswithhomogeneousfluxesgoesbackto thework of Bardos, leRoux andNedelec [3], who considered whend

=

1 solutions in L^∞_t

(

BVx

)

, whichin view of the BV-regularity have traces; see also Dubois and Le Floch [4]. Then Otto [7,8] introduced the notion of boundary entropies and used weak traces, whose existence is a simpleconsequence of theKruzkov definition,to studysolutions withboundary data in h

∈

^L∞

()

withoutassuming the existenceof (strong)traces.LaterPorretta andVovelle[10] andAmmar, Carrilloand Wittbold [1] extended thetheory to L¹-datausing thenotion of renormalizedsolutions. Ifone assumes that the flux is genuinely nonlinear, it is possible to bypass the need of boundary entropies, since, inview of a result ofVasseur [12], entropy solutionsadmit inthiscasestrongtraces.Theresultof [12] was generalizedbyPanov [9].Using [9], Andreianov andSbihi [2] studiedboundaryvalueproblemswithavarietyofboundaryconditions.

Weintroducenext anotionofentropysolutions,whichdoesnotrelyonanykindofstrongtraces,extendstoinhomo- geneousfluxesandyieldsarathertransparentuniquenessproofthatdoesnotrequireanyboundaryentropies.

Forsimplicity,weassumethath

≡

^{0 andwereferto [6] forthegeneralcase.Moreover,ifg}

:

^U

→

R^,g

(

x

)δ

∂U standsfor thedistribution,definedforall

φ ∈

D

(

U

)

by

∂Ug

(

y

)φ (

y

)

dS

(

y

)

.

Definition3.1.Afunctionu

∈

^C

( [

⁰

,

T

];

^L1

(

U

)) ∩

^L∞

(

U

× (

0

,

T

))

isanentropysub- (resp.super-)solutionto(16) withh

≡

⁰ if,foreveryk

∈

R^,

(

u

∨

^k

)

t

+

^divxf

(

u

∨

^k

) −

¹_{u<k}div_xf

(

k

,

x

) ≤

^{0 in} D

(

U

× (

0

,

T

)),

(resp.

(

u

∧

k

)

t

+

divxf

(

u

∧

k

,

x

) −

1_{u>k}divxf

(

k

,

x

) ≥

0 in D

(

U

× (

0

,

T

)),

and,foreveryk

≥

^{0 (resp.k}

≤

^0),

(

u

∨

^k

)

t

+

^divxf

(

u

∨

^k

) −

¹{u<k}divxf

(

k

,

x

) +

^f

(

k

,

x

) · ν (

x

)δ

_∂U

≤

^{0 in} D

(

U

× (

0

,

T

)),

(17)

(

resp.

(

u

∧

^k

)

t

+

^divxf

(

u

∧

^k

) −

¹_{u>k}div_xf

(

k

,

x

) +

^f

(

k

,

x

) · ν (

x

)δ

∂U

≥

^{0 in} D

(

U

× (

0

,

T

))).

(18) Afunction u

∈

^C

( [

⁰

,

T

];

^L1

(

U

)) ∩

^L∞

(

U

× (

0

,

T

))

is an entropysolution to(16) withh

≡

^{0 if itis bothentropy sub- and} super-solution.

The meaning of (17) (resp. (18)) isthat, for every k

≥

^{0 (resp. k}

≤

^{0) and any}non-negative andsmooth

φ

, whichis compactlysupportedinU andinD

((

0

,

T

))

,

d dt

U

(

u

(

x

,

t

) ∨

^k

)φ (

x

)

dx

−

U

f

(

u

∨

^k

) ·

^D

φ (

x

)

dx

−

U

1_{u<k}div_xf

(

k

,

x

)φ (

x

)

dx

+

∂U

f

(

k

,

y

) · ν (

y

)φ (

y

)

dS

(

y

) ≤

⁰

,

(resp.

d dt

U

(

u

(

x

,

t

) ∧

^k

)φ (

x

)

dx

−

U

f

(

u

∧

^k

) ·

^D

φ (

x

)

dx

−

U

1_{u>k}div_xf

(

k

,

x

)φ (

x

)

dx

+

∂U

f

(

k

,

y

) · ν (

y

)φ (

y

),

dS

(

y

) ≥

⁰

.)

We remarkthat (17) (resp.(18)) isequivalent tosaying that, forevery k

≥

^{0 (resp.k}

≤

^{0)) andany}non-negativeand smooth

φ

,whichiscompactlysupportedinU,

d dt

U

(

u

(

x

,

t

) −

^k

)

₊

φ (

x

)

dx

−

U

[

^f

(

u

(

x

,

t

),

x

) −

^f

(

k

,

x

)] ·

^D

φ (

x

)

1_{u≥k}dx

+

U

1_{u≥k}div_xf

(

k

,

x

)φ (

x

)

dx

≤

⁰

,

(resp.

d dt

U

(

k

−

^u

(

x

,

t

))

₊

φ (

x

)

dx

−

U

[

^f

(

k

,

x

) −

^f

(

u

(

x

,

t

),

x

) ] ·

^D

φ (

x

)

1_{u≤k}dx

+

U

1_{u≤^k}divxf

(

k

,

x

)φ (

x

)

dx

≤

0

).

NextwestatethebasiccomparisonandexistenceresultforsolutionsinCt

(

L¹

)) ∩

^L∞x,t.Wediscussmoregeneralresults in [6].

Theorem3.2.Assume(2)withU inplaceofR^dandletu

,

v

∈

^C

([

⁰

,

T

];

^L1

(

U

) ∩

^L∞

(

U

× (

0

,

T

))

berespectivelysub- andsuper- solutionsto(16)withh

≡

^{0.Then,foralls}

,

t

∈ (

0

,

T

)

suchthats

<

t,

U

|

u

(

x

,

t

) −

v

(

x

,

t

)|

dx

≤

U

|

u

(

x

,

s

) −

v

(

x

,

s

)|

dx

.

If,inaddition,(8)and(11)withU inplaceofR^dholdandu0

∈ (

L¹

∩

^L∞

)(

U

)

,then(16)withh

≡

^{0hasasolutioninC}

([

⁰

,

T

];

^U

) ∩

L^∞

(

U

× (

0

,

T

))

.