COMPENSATED COMPACTNESS FOR 2D CONSERVATION LAWS

Description

Niveau: Supérieur, Doctorat, Bac+8
COMPENSATED COMPACTNESS FOR 2D CONSERVATION LAWS EITAN TADMOR, MICHEL RASCLE, AND PATRIZIA BAGNERINI Contents 1. Introduction and statement of main results 1 2. Strong convergence – a single entropy suffices in the 1D case 3 3. Strong convergence – two entropies suffice in the 2D case 4 3.1. Compensated compactness in 2D conservation laws 5 3.2. 2D examples 8 4. Kinetic formulation – the multidimensional case 9 5. Convergence of multidimensional finite volume schemes 11 6. Appendix 13 References 13 Abstract. We introduce a new framework for studying two-dimensional conservation laws by compensated compactness arguments. Our main result deals with 2D conserva- tion laws which are nonlinear in the sense that their velocity fields are a.e. not co-linear. We prove that if u? is a family of uniformly bounded approximate solutions of such equa- tions with H?1-compact entropy production and with (a minimal amount of) uniform time regularity, then (a subsequence of) u? convergences strongly to a weak solution. We note that no translation invariance in space — and in particular, no spatial regularity of u(·, t) is required. Our new approach avoids the use of a large family of entropies; by a judicious choice of entropies, we show that only two entropy production bounds will suffice. We demonstrate these convergence results in the context of vanishing viscosity, kinetic BGK and finite volume approximations.

• strong convergence


• multidimensional problems


• compact entropy


• u?


• multi-dimensional nonlinearity


• compensated compactness


• h?1loc


• volume schemes


• bv-bounds

Sujets

• Strong convergence

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 10
• Langue: English

Extrait

COMPENSATEDCOMPACTNESSFOR2DCONSERVATIONLAWS EITANTADMOR,MICHELRASCLE,ANDPATRIZIABAGNERINI Contents 1.Introductionandstatementofmainresults 2.Strongconvergence–asingleentropysufficesinthe1Dcase 3.Strongconvergence–twoentropiessufficeinthe2Dcase 3.1.Compensatedcompactnessin2Dconservationlaws 3.2.2Dexamples 4.Kineticformulation–themultidimensionalcase 5.Convergenceofmultidimensionalfinitevolumeschemes 6.Appendix References 1 3 4 5 8 9 11 31 31 Abstract.Weintroduceanewframeworkforstudyingtwo-dimensionalconservation lawsbycompensatedcompactnessarguments.Ourmainresultdealswith2Dconserva- tionlawswhichare nonlinear inthesensethattheirvelocityfieldsarea.e.notco-linear. Weprovethatif uε isafamilyofuniformlyboundedapproximatesolutionsofsuchequa- tionswith H − 1 -compactentropyproductionandwith(aminimalamountof)uniform timeregularity,then(asubsequenceof) uε convergencesstronglytoaweaksolution.We notethatnotranslationinvarianceinspace—andinparticular,nospatialregularity of u (
,t )isrequired.Ournewapproachavoidstheuseofalargefamilyofentropies;by ajudiciouschoiceofentropies,weshowthatonly two entropyproductionboundswill suffice.Wedemonstratetheseconvergenceresultsinthecontextofvanishingviscosity, kineticBGKandfinitevolumeapproximations.Finally,theintimateconnectionbe- tweenour2Dcompensatedcompactnessargumentsandthenotionofmulti-dimensional nonlinearitybasedonkineticformulationisclarified. KeyWords: Conservationlaws,entropybounds,compensatedcompactness,kineticfor- mulation. AMSsubjectclassification: Primary35L65,76P05;Secondary65M12,65M60. 1.Introductionandstatementofmainresults Currently,therearefourmainapproachestostudytheexistenceofsolutionsforquasi- linearhyperbolicconservationlaws.Firstwasthestandardtoolofcompactnessbased onaprioriBVbounds.Then,fromthemid-eightiesthroughthemid-nineties,theother threeapproachesofcompensatedcompactness,measurevaluedsolutionsandkineticfor- mulationsweredeveloped,allofwhichappealtoapriorientropyproductionbounds. CompactnessargumentsbasedonBVboundswereprovenasthemosteffectivetoolfor studyinggeneralone-dimensionalsystemsofconservationlaws.Thelonglineofresultsin November25,2004 1