Regularity in kinetic formulations via averaging lemmas

Pierre-Emmanuel Jabin

Description

Regularity in kinetic formulations via averaging lemmas Pierre-Emmanuel Jabin and Benoıt Perthame email: , Ecole Normale Superieure Departement de Mathematiques et Applications, CNRS UMR 8553 45 rue d'Ulm, 75230 Paris Cedex 05, France in Memory of J.-L. Lions Abstract. We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like ? = 3 in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to use velocity regularity for the solution to the transport equation under consideration. The method of proof is based on a decomposition of the density in Fourier space, combined with the K-method of real interpolation. Key words. Regularizing effects, kinetic formulation, averaging lemmas, hyperbolic equations, line-energy Ginzburg-Landau. 1 Introduction Kinetic formulations allow to consider nonlinear problems (balance laws or variational problems) and, using a nonlinear function f of the unknown, to 1

• can also


• ???lr∞ ≤


• transport equation


• regularizing effects


• also optimal


• kinetic formulation


• standard averaging

Sujets

• Generic scalar transport equation

Informations

• Publié par: pefav
• Nombre de lectures: 9
• Langue: English

Extrait

Regularityinkineticformulationsvia averaginglemmas Pierre-EmmanuelJabinandBenoıˆtPerthame email:[e-mail],[e-mail] E ´ coleNormaleSupe´rieure De´partementdeMathe´matiquesetApplications,CNRSUMR8553 45rued’Ulm,75230ParisCedex05,France inMemoryofJ.-L.Lions Abstract.Wepresentanewclassofaveraginglemmasdirectlymotivated bythequestionofregularityfordifferentnonlinearequationsorvariational problemswhichadmitakineticformulation.Inparticulartheyimprovethe knownregularityforsystemslikeγ=3inisentropicgasdynamicsorinsome variationalproblemsarisinginthinmicromagneticfilms.Theyalsoallowto obtaindirectlythebestknownregularizingeffectinmultidimensionalscalar conservationlaws. Thenewingredienthereistousevelocityregularityforthesolutiontothe transportequationunderconsideration.Themethodofproofisbasedona decompositionofthedensityinFourierspace,combinedwiththeK-method ofrealinterpolation. Keywords.Regularizingeffects,kineticformulation,averaginglemmas, hyperbolicequations,line-energyGinzburg-Landau. 1Introduction Kineticformulationsallowtoconsidernonlinearproblems(balancelawsor variationalproblems)and,usinganonlinearfunctionfoftheunknown,to 1

transformtheseproblemsinasingularlineartransportequationonf.The simplestexampleisthatoftheentropysolutionu ∈ C( R + ;L 1 ( R d ))toa multidimensionalscalarconservationlaw d ∂ t u(t,x)+divA(u)=0,t>0,x ∈ R , (1.1) ∂ t S(u(t,x))+divη S (u) ≤ 0, f R orallconvexfunctionS( ∙ )withS(0)=0andusingthenotationsη S (u)= 0 u S 0 ( ∙ )a( ∙ ),a=A 0 : R → R d .Then,wedefine,forv ∈ R ,the‘equilibrium’ functionf(t,x,v)thanksto   +1,for0<v<u(t,x), f(t,x,v)= − 1,foru(t,x)<v<0,(1.2)  0,otherwise. Thetheoryofkineticformulationsstatesthat(1.1)isequivalenttowritethe kineticequationonf ∂ t f+a(v) ∙r x f=∂ v m(t,x,v),(1.3) forsomeunknownnonnegativeboundedmeasurem.Thederivationisob- tainedbyintegrating(1.3)againstS 0 (v),andsincewehave ZZ S(u)=S 0 (v)f(t,x,v)dv,η S (u)=S 0 (v)a(v)f(t,x,v)dv, RR thekineticformulationturnsouttoprovidetheinequalities Z ∂ t S(u(t,x))+divη S (u)= − S 00 (v)m(t,x,v)dv. R Thereforetheinequalitiesinthesecondequationof(1.1)areequivalenttothe positivityofm.Alsoacontrolofthetotalmassofthemeasureisobtained usingS(v)=v 2 /2intheaboveequality ∞ ZZ 1 m(t,x,v)dtdvdx ≤k u 0 k 2 L 2 ( R d ) . 0 R × R d 2 Thismethodturnsouttoprovideatoolforstudyingregularizingeffects forthehyperbolicequation(1.1),whenanon-degeneracyconditiononthe fluxesAissatisfied.Indeed,averaginglemmasmaybeappliedtothelinear 2