Hypocoercive diffusion operators Cedric Villani

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Hypocoercive diffusion operators Cedric Villani Abstract. In many problems coming from mathematical physics, the association of a degenerate diffusion operator with a conservative operator may lead to dissipation in all variables and convergence to equilibrium. One can draw an analogy with the well-studied phenomenon of hypoellipticity in regularity theory, and actually both phenomena have been studied together. Now a distinctive theory of “hypocoercivity” is starting to emerge, with already some striking results, and several challenging open problems. Mathematics Subject Classification (2000). Primary 35B40; Secondary 35K70, 76P05. Keywords. Hypocoercivity, hypoellipticity, diffusion equations, spectral gap, logarith- mic Sobolev inequalities, Fokker–Planck and Boltzmann equations, H Theorem. Introduction During the past decade, considerable progress has been achieved in the qualitative study of diffusion equations in large time, be it for linear or nonlinear models. Quantitative functional methods have become especially popular. Here are some of the keywords in the field: spectral gap (Poincare) inequalities, logarithmic Sobolev inequalities, analysis of entropy production, gradient flows, rescalings. Most of the time, estimates on the rate of convergence are established in the end by means of some Gronwall-type inequality dE/dt ≤ ??(E), where E is a Lyapunov functional for the system. Among a large literature, I shall only quote some of my own works: entropy production estimates for the spatially homogeneous Boltzmann equation, in collaboration with Giuseppe Toscani [25], [26], [28]; and for certain nonlinear diffusion equations with a convex mean

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HypocoercivediﬀusionoperatorsCe´dricVillaniAbstract.Inmanyproblemscomingfrommathematicalphysics,theassociationofadegeneratediﬀusionoperatorwithaconservativeoperatormayleadtodissipationinallvariablesandconvergencetoequilibrium.Onecandrawananalogywiththewell-studiedphenomenonofhypoellipticityinregularitytheory,andactuallybothphenomenahavebeenstudiedtogether.Nowadistinctivetheoryof“hypocoercivity”isstartingtoemerge,withalreadysomestrikingresults,andseveralchallengingopenproblems.MathematicsSubjectClassiﬁcation(2000).Primary35B40;Secondary35K70,76P05.Keywords.Hypocoercivity,hypoellipticity,diﬀusionequations,spectralgap,logarith-micSobolevinequalities,Fokker–PlanckandBoltzmannequations,HTheorem.IntroductionDuringthepastdecade,considerableprogresshasbeenachievedinthequalitativestudyofdiﬀusionequationsinlargetime,beitforlinearornonlinearmodels.Quantitativefunctionalmethodshavebecomeespeciallypopular.Herearesomeofthekeywordsintheﬁeld:spectralgap(Poincare´)inequalities,logarithmicSobolevinequalities,analysisofentropyproduction,gradientﬂows,rescalings.Mostofthetime,estimatesontherateofconvergenceareestablishedintheendbymeansofsomeGronwall-typeinequalitydE/dt≤−Φ(E),whereEisaLyapunovfunctionalforthesystem.Amongalargeliterature,Ishallonlyquotesomeofmyownworks:entropyproductionestimatesforthespatiallyhomogeneousBoltzmannequation,incollaborationwithGiuseppeToscani[25],[26],[28];andforcertainnonlineardiﬀusionequationswithaconvexmean-ﬁeldinteraction,incollaborationwithJose´AntonioCarrilloandRobertMcCann[2].Whilethesesubjectsarestillveryactive,inthistextIshallfocusonanewerdirectionofresearchwhichhasemergedonlyafewyearsago,andcanbelooselydescribedas“theroleofthenon-dissipativepartinthedissipationprocess”.Indeed,ithappensnotsorarelythatthedissipativepropertiesofanequationarestronglyinﬂuencedbysomeoftheconservativetermsinthisequation.Thisstatementinitselfisnothingnew,sinceitisalmostobviousinthecontextofhydro-dynamics(dissipativityinNavier–Stokesiscertainlyconsiderablymorecomplexthanintheheatequation).Inthecontextofdiﬀusionequations,theinteractionbetweendissipativeandconservativetermsisalsowell-known,sinceitisattheba-

sisofthephenomenonofhypoellipticity.Tomakethediscussionabitmoreprecise,letmerecallaparticularlysimpletheoremofhypoellipticregularization,whichisadirectconsequenceofLarsHo¨rmander’scelebratedregularitytheorem[20].LetA1,...,AkandPBbeC∞vectorﬁeldsonRN,identiﬁedwithderivationoperators,andletL=−Aj2+B.Iftherankof(A1,...,Ak)isstrictlylessthanN,thentheoperatorLisnotelliptic,andthereisnoapriorPireasonwhyPthesemigroupe−tLwouldberegularizinginallvariables.Butif−[Aj,B]2−Aj2iselliptic,where[Aj,B]istheLiebracketbetweenAjandB,thene−tLisregularizinginallvariables,andtheoperatorLissaidtobehypoelliptic.(Thisisnottheclassicaldeﬁnitionofhypoellipticity,butitwilldoforthepurposeofthispresentation.)Weseeherehowthe“nondissipative”ﬁrst-orderoperatorBinteractswiththe“dissipativepart”ofL,ormorepreciselythederivationoperatorsAj,toproducethemissingdirectionsofregularization.PossiblythemostimportantinstanceofapplicationistotheoperatorL=−Δv+v∇x,where(x,v)∈Rn×Rn;inthatcaseAj=∂/∂vj,B=v∇x,[Aj,B]=∂/∂xj.Thecorrespondingevolutionequation∂tf+Lf=0isdegenerate,butstillpresentssomeofthetypicalfeaturesofaparabolicequation;theword“ultraparabolic”issometimesusedforit.Hypoellipticregularityhasbeentheobjectofhundredsofworksforthepastfourdecades.Butwhatwasunderstoodonlyveryrecentlyisthatquitesimilarphenomenaariseinthestudyofratesofconvergencetoequilibrium.Todescribethis,Ishallusetheword“hypocoercivity”,whichwassuggestedtomebyThierryGallay.AtypicalhypocoercivitytheoremwillgivesuﬃcientconditionsonanoperatorLsothate−tLwillconvergetoequilibriumatacertainrate,eventhoughLisnot“coercive”,inthesensethatthekernelofitsdissipativepartismuchlargerthanthesetofequilibria.Hypoellipticityandhypocoercivityareoftenfoundtogether,andhavebeenac-tuallystudiedtogether,byreﬁnedhypoelliptictechniques[6],[7],[15],[16],[19],andsometimesbyprobabilisticmethods[8],[22],[23].However,thesetwophe-nomenaaredistinct:Eachofthemcanoccurwithouttheother;andthestructureswhichunderliethemarenotexactlythesame.Thismotivatesthedevelopmentofaseparatetheoryofhypocoercivity.Inthesequel,Ishallpresentsomeoftheﬁrstresultsinthisdirection.Acknowledgement.Theideasexposedinthesequelhavebeneﬁtedfrominter-actionswithmanypeoplewhoarequotedwithinthetext.WarmthanksareduetoMartinHairer,Fre´de´ricHe´rauandCle´mentMouhotfortheirdetailedcommentsonapreliminaryversionofthesenotes;andtoThierryGallayforilluminatingdiscussions.1.MotivationsInthissectionIshalldescribesomeconcreteexampleswhichmotivatethestudyofhypocoercivity.Allofthemcomefrommathematicalphysics,andnoneofthemisacademic.Ofcoursethelistisfarfromexhaustive.2

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