On some properties of kinetic and hydrodynamics eq

OnsomepropertiesofkineticandhydrodynamicequationsforinelasticinteractionsA.V.Bobylev,J.A.Carrilloy,I.GambazMay6,1999AbstractWeinvestigateaBoltzmannequationforinelasticscatteringinwhichtherela-tivevelocityinthecollisionfrequencyisapproximatedbythethermalspeed.Theinelasticityisgivenbyavelocityvariablerestitutioncoecient.ThisequationistheanalogoustotheBoltzmannclassicalequationforMaxwellianmolecules.WestudythehomogeneousregimeusingFourieranalysismethods.Weanalyzetheexistenceanduniquenessquestions,linearizedoperatoraroundDiracdeltafunc-tion,self-similarsolutionsandmomentequations.Weclarifytheconditionsunderwhichtheself-similarsolutionsdescribethelargetimeasymptoticbehaviorofthesystem.Weobtainformallyahydrodynamicdescriptionfornearelasticparticlesundertheassumptionofconstantandvariablerestitutioncoecient.Wedescribethelinearlong-wavestability/instabilityforthehomogeneouscoolingstates.Keywords:granularmedia,Boltzmannequation,inelasticcollisions,homogeneoussolutions,dissipativehydrodynamics,stability.1IntroductionTheaimofthispaperistoclarifysomequestionsconcerningprincipalpropertiesofkineticequationsforgranularmedia.Weusethewell-knownandwide-acceptedmodelgivenbythegeneralizedBoltzmann-Enskogequationforadensegasofinelasticspheresasthebasisofourstudy.Forthesakeofreader’sconveniencewedescribeabriefschemeofitsderivationinSection2.M.V.KeldyshInstituteofAppliedMathematics,RussianAcademyofSciences,Moscow,RUSSIA.yDepartmentofMathematics,UniversityofTexasatAustin,78712Austin-Texas,USA;onleavefromDepartamentodeMatematicaAplicada,FacultaddeCiencias,UniversidaddeGranada,18071Granada,SPAIN.zDepartmentofMathematics,UniversityofTexasatAustin,78712Austin-Texas,USA.1Weuseavelocitydependentrestitutioncoecientwhichcharacterizestheinelasticityofcollisions.Theconstantrestitutioncoecientcaseleadstowell-knownunrealisticphysicalstates.Infact,therestitutioncoecientmaydependonrelativevelocityinsuchawaythatcollisionswithsmallrelativevelocityareclosetobeelastic.Thistypeofrestitutioncoecientismorerealisticandithasbeenusedinmoleculardynamicssimulationofoscillatedgranularmedia[5].Besides,ithasbeenprovedthatadynamicalmodelwiththeconstantcoecientleadstoinelasticcollapse[2,10].Ontheotherhand,thecollapsedoesnotoccurforvariablerestitutioncoecientwithappropriatebehaviorforsmallvaluesofrelativespeed[13].Dealingwithdensegasesorgranularmediawealwaysassume(directlyorindirectly)thatthemeanfreepathisrelativelysmallsinceweareoutsideoftheBoltzmann-Gradlimit.Thisimpliesatransitiontoacertainhydrodynamicregime.Attheformal(physi-cal)levelofdescriptionthistransitioncanbemadebydierentways(Gradmethod[15],Chapman-Enskogexpansion[14],etc).However,inallcases,weneedtoanswerthefollowingquestions:(a)howtodescribelarge-timeasymptoticsofsolutionstothespatiallyhomogeneousequation;and(b)whatcanbesaidaboutmainpropertiesofcorrespondinglimitequations(hydrodynamics).Ourworkcanbeconsideredasanattempttopartlyclarifythesequestions,especiallyinthecaseofvariablerestitutioncoecient.Inparticular,Sections3-6aredevotedtothespatiallyhomogeneousproblemandSection7-8tosomestability/instabilitypropertiesofthelimitdissipativeEulerequations.Anextensivesurveyofphysicalliteraturecanbefoundin[12].Theexistencetheoryfortheinelastichard-sphereBoltzmann-Enskogequationwasanalyzedin[11].Adetailtheoryofone-dimensionalgranularowswasrecentlydevelopedin[3,4].Theseandothercitedpublicationsareonlyasmallpartofahugeliteraturerelatedtoowsofinelasticgranularmaterials.Akeyideaofourapproachistouseasimplied(pseudo-maxwellian)versionoftheBoltzmann-Enskogequation.PhysicalreasonsforsuchasimplicationaredescribedindetailattheendofSection2.Thissimplicationisespeciallyeectiveinthespatiallyhomogeneouscase.InordertoanalyzethisspatiallyhomogeneousequationweusetheFouriertransformmethodappliedintheBoltzmannequationforMaxwellianmoleculesbyoneoftheauthors[6,7].Here,themaindierenceistheabsenceofconservationofenergyatthelevelofthecollisionmechanism,andthus,attheleveloftheBoltzmannequation.Infact,aMaxwelliandistributioncannotbeasolutionoftheinelasticBoltzmannequation.However,theDiracdeltadistributionisobviouslyinthekernelofthecollisionoperator.Therefore,withtheFouriermethod,wecandescribethespectrumofthelinearizedhomogeneousequationaroundtheDiracdeltadistribution.Inaddition,weclarifytheexistenceofself-similarsolutionsforthismodel.Thistypeofsolutionshavebeenconsideredbeforefortheinelastichard-sphereBoltzmannequation[8,14].Theyhavebeencalledhomogeneouscoolingstates.Thenearlyelasticcasewasstudiedin[14].Inthepresentmodelwestateandprovethepreciseconditionsunder2whichtheseself-similarsolutionsexist,andthendescribethelarge-timebehaviorofthesystem.TheseresultsareaccomplishedusingeigenfunctionexpansionofthesolutionsintheFouriertransformequation.Finally,westudythemomentequationsforthissystem.Weanalyzetwocasesunderwhichtheself-similarsolutionsareasymptoticallyrelevant.Thesecasesare:a)nearlyelasticparticleswithconstantorvariablerestitutioncoecientandb)smalltemperature.Inbothcases,theself-similarsolutionisnearaMaxwelliandistri