An Interesting Class of Partial Differential Equat

arXiv:0707.3708v2[math-ph]28Aug2007ANINTERESTINGCLASSOFPARTIALDIFFERENTIALEQUATIONSWEN-ANYONGAbstract.Thispaperpresentsanobservationthatunderreasonableconditions,manypartialdifferentialequationsfrommathematicalphysicspossessthreestructuralproper-ties.OneofthemcanbeunderstandasavariantofthecelebratedOnsagerreciprocalrelationinModernThermodynamics.Itdisplaysadirectrelationofirreversibleprocessestotheentropychange.Weshowthatthepropertiesimplyvariousentropydissipationconditionsforhyperbolicrelaxationproblems.Asanapplicationoftheobservation,weproposeanapproximationmethodtosolverelaxationproblems.Moreover,theobser-vationisinterpretedphysicallyandverifiedwitheight(setsof)systemsfromdifferentfields.Contents1.Introduction12.AnObservation33.AnApproximationMethod64.PhysicalInterpretations85.FourSpecificExamples96.RadiationHydrodynamics127.ChemicallyReactiveFlows138.MomentClosureSystems179.DiscreteVelocityModels20References211.IntroductionThegoalofthispaperistodrawattentiontoaclassofpartialdifferentialequations(PDEs)oftheform(1.1)Ut+dXj=1Fj(U)xj=Q(U).HereUistheunknownn-vector-valuedfunctionof(x,t)≡(x1,x2,···,xd,t)∈Rd×[0,+∞),takingvaluesinanopensubsetGofRn(calledstatespace);Q(U)andFj(U)(j=1,2,···,d)aregivenn-vector-valuedsmoothfunctionsofU∈G;andthesubscriptstandxjrefertothepartialderivativeswithrespecttotandxj,respectively.AsfundamentalPDEsandasintermediatemodels[7,15,19]betweentheBoltzmannequation[2]andhyperbolicconservationlaws[4],systemsoffirst-orderPDEswithsourcetermsdescribevariousirreversibleprocessesofscalartype[11].Importantexamplesoccur12W.-A.Yonginchemicallyreactiveflows[9],radiationhydrodynamics[17,22],invisicidgasdynamicswithrelaxation[31],nonlinearoptics[12],andsoon.Sincethelastdecade,PDEsoftheform(1.1)haveattractedmuchattention.See[20,28,25]andreferencescitedtherein.Oneofthemaininterestsistoidentifyasetofstructuralproperties(axioms)thataresatisfiedbymostofimportantequationsfromapplicationsand,meanwhile,provideaconvenientframeworkforthedevelopmentofmathematicaltheories.Inthisregard,twostabilityconditionsandvariousentropydissipationconditionshavebeenproposedin[26]and[3,19,29,13,25],respectively.Seealso[28,23].Allthoseconditionsaregeneralizationsofthewell-knownsubcharacteristiccondition[16]for(1.1)withn=2andd=1.For(1.1),suchaconditionisthesameinspiritastheH-theoremfortheBoltzmannequation[2]andastheentropyconditionforconservationlaws[4].Inthispaper,wepresentanobservationthatunderreasonableassumptions,manyequationsoftheform(1.1)frommathematicalphysicsfallwithinaclasscharacterizedwiththefollowingthreeproperties.(I)Everysystemintheclassadmitsastrictlyconvexentropyfunction[10,6],(II)thesourcetermcanbewrittenasaproductofanon-positivesymmetricmatrixandthecorrespondingentropyvariable,and(III)thesymmetricmatrixhasaconstantnull-space.Thefirstpropertyisthewell-knownentropyconditionforconservationlawsandcor-respondstotheclassicalprinciplesofthermodynamics.Property(II)canbeunderstandasavariantofthecelebratedOnsagerreciprocalrelationinModernThermodynamics[11,14]andimpliesthesecondlawofthermodynamics.Itdisplaysadirectrelationofirreversibleprocessestotheentropychange.Property(III)expressesthefactthatphysi-callawsofconservationholdtrue,nomatterwhatstatetheunderlyingthermodynamicalsystemisin(equilibrium,non-equilibrium,andsoon).Wewillverifythethreepropertiesforeight(setsof)systemsoftheform(1.1)arisingingasdynamicswithdampingorwithrelaxation,nonlinearoptics,radiationhydrody-namics,chemicalreactions,kinetictheories(bothmomentclosuresystemsanddiscretevelocitymodels),andsoon.Furthermore,weshowthatthepropertiesensureauniquelydefinedMaxwellianandimplyvariousentropydissipationconditionsintheliteratureforhyperbolicrelaxationproblems.Thus,allthegeneralresultsin[27,30,18,23,29,13,25,5]applytotheaforementionedfields.WenoticethattheexamplesinSections8and9haveacommonkineticorigin.More-over,weknowfrom[8]thatthechemicalsystemsinSection7haveasimilarorigin.ItwouldbeinterestingtoincludetheradiativegasexampleofSection6inthesamebasket,withtheideathatradiationisassociatedwithparticles(photons)collisions.Inotherwords,weshowthatthekinetictheoryyieldstheOnsagerrelationforgasmixtures.Asanapplicationofourobservation,weproposeanapproximationmethodtosolvetherelaxationproblems.Theaccuracyofthemethodisanalysedforinitialvalueproblemswithsmoothinitialdata,byusingtheresultsin[27].Inthisanalysis,animportantingredientisacontinuationprincipleforhyperbolicsingularlimitproblems(Lemma9.1in[28]andtheappendixin[1]).Furtheranalysisandapplicationsoftheapproximationmethodaredesirable.Ourdiscussionsindicatethattheabovethreepropertieshaveasolidbasis,frombothmathematicalandphysicalpointsofview.Thus,itseemsreasonabletotakethepropertiesasrequirementsinconstructionofnewmathematicalmodelsforirreversiblephenomena.Aninterestingclassofpartialdifferentialequations3Thepaperisorganizedasfollows.InSection2wepresenttheobservationanddiscussitsmathematicalconsequences.Section3isdevotedtotheapproximationmethod.SomephysicalinterpretationsaregiveninSection4.Therestofthepaperdealswiththeeight(setsof)exa