Hypocoercive diffusion operators Cedric Villani

Cédric Villani

Description

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

• diffusion equations


• mentioned hilbert space


• distribution has


• many models


• hypocoercive diffusion operators


• fokker–planck equation


• equations satisfied


• interaction between

Sujets

• Carrillo
• McCann
• Villani
• Toscano

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 61
• Langue: English

Extrait

Hypocoercivediffusionoperators Ce´dricVillani Abstract.Inmanyproblemscomingfrommathematicalphysics,theassociationofa degeneratediffusionoperatorwithaconservativeoperatormayleadtodissipationinall variablesandconvergencetoequilibrium.Onecandrawananalogywiththewell-studied phenomenonofhypoellipticityinregularitytheory,andactuallybothphenomenahave beenstudiedtogether.Nowadistinctivetheoryof“hypocoercivity”isstartingtoemerge, withalreadysomestrikingresults,andseveralchallengingopenproblems. MathematicsSubjectClassification(2000).Primary35B40;Secondary35K70, 76P05. Keywords.Hypocoercivity,hypoellipticity,diffusionequations,spectralgap,logarith- micSobolevinequalities,Fokker–PlanckandBoltzmannequations, H Theorem. Introduction Duringthepastdecade,considerableprogresshasbeenachievedinthequalitative studyofdiffusionequationsinlargetime,beitforlinearornonlinearmodels. Quantitativefunctionalmethodshavebecomeespeciallypopular.Herearesomeof thekeywordsinthefield:spectralgap(Poincare´)inequalities,logarithmicSobolev inequalities,analysisofentropyproduction,gradientflows,rescalings.Mostofthe time,estimatesontherateofconvergenceareestablishedintheendbymeansof someGronwall-typeinequalitydE/dt≤−Φ(E),whereEisaLyapunovfunctional forthesystem.Amongalargeliterature,Ishallonlyquotesomeofmyownworks: entropyproductionestimatesforthespatiallyhomogeneousBoltzmannequation, incollaborationwithGiuseppeToscani[25],[26],[28];andforcertainnonlinear diffusionequationswithaconvexmean-fieldinteraction,incollaborationwithJose´ AntonioCarrilloandRobertMcCann[2]. Whilethesesubjectsarestillveryactive,inthistextIshallfocusonanewer directionofresearchwhichhasemergedonlyafewyearsago,andcanbeloosely describedas“theroleofthenon-dissipativepartinthedissipationprocess”. Indeed,ithappensnotsorarelythatthedissipativepropertiesofanequation arestronglyinfluencedbysomeoftheconservativetermsinthisequation.This statementinitselfisnothingnew,sinceitisalmostobviousinthecontextofhydro- dynamics(dissipativityinNavier–Stokesiscertainlyconsiderablymorecomplex thanintheheatequation).Inthecontextofdiffusionequations,theinteraction betweendissipativeandconservativetermsisalsowell-known,sinceitisattheba-

sisofthephenomenonofhypoellipticity.Tomakethediscussionabitmoreprecise, letmerecallaparticularlysimpletheoremofhypoellipticregularization,whichis adirectconsequenceofLarsHo¨rmander’scelebratedregularitytheorem[20].Let A 1 ,...,A k and P BbeC ∞ vectorfieldson R N ,identifiedwithderivationoperators, andletL=−A j 2 +B.Iftherankof(A 1 ,...,A k )isstrictlylessthanN,then theoperatorLisnotelliptic,andthereisnoaprior P ireasonwhy P thesemigroup e −tL wouldberegularizinginallvariables.Butif−[A j ,B] 2 −A j 2 iselliptic, where[A j ,B]istheLiebracketbetweenA j andB,thene −tL isregularizinginall variables,andtheoperatorLissaidtobehypoelliptic.(Thisisnottheclassical definitionofhypoellipticity,butitwilldoforthepurposeofthispresentation.) Weseeherehowthe“nondissipative”first-orderoperatorBinteractswiththe “dissipativepart”ofL,ormorepreciselythederivationoperatorsA j ,toproduce themissingdirectionsofregularization.Possiblythemostimportantinstanceof applicationistotheoperatorL=−Δ v +v
∇ x ,where(x,v)∈ R n × R n ;inthat caseA j =∂/∂v j ,B=v
∇ x ,[A j ,B]=∂/∂x j .Thecorrespondingevolution equation∂ t f+Lf=0isdegenerate,butstillpresentssomeofthetypicalfeatures ofaparabolicequation;theword“ultraparabolic”issometimesusedforit. Hypoellipticregularityhasbeentheobjectofhundredsofworksforthepast fourdecades.Butwhatwasunderstoodonlyveryrecentlyisthatquitesimilar phenomenaariseinthestudyofratesofconvergencetoequilibrium.Todescribe this,Ishallusetheword“hypocoercivity”,whichwassuggestedtomebyThierry Gallay.Atypicalhypocoercivitytheoremwillgivesufficientconditionsonan operatorLsothate −tL willconvergetoequilibriumatacertainrate,eventhough Lisnot“coercive”,inthesensethatthekernelofitsdissipativepartismuch largerthanthesetofequilibria. Hypoellipticityandhypocoercivityareoftenfoundtogether,andhavebeenac- tuallystudiedtogether,byrefinedhypoelliptictechniques[6],[7],[15],[16],[19], andsometimesbyprobabilisticmethods[8],[22],[23].However,thesetwophe- nomenaaredistinct:Eachofthemcanoccurwithouttheother;andthestructures whichunderliethemarenotexactlythesame.Thismotivatesthedevelopmentof aseparatetheoryofhypocoercivity.Inthesequel,Ishallpresentsomeofthefirst resultsinthisdirection. Acknowledgement. Theideasexposedinthesequelhavebenefitedfrominter- actionswithmanypeoplewhoarequotedwithinthetext.Warmthanksaredueto MartinHairer,Fre´de´ricHe´rauandCle´mentMouhotfortheirdetailedcomments onapreliminaryversionofthesenotes;andtoThierryGallayforilluminating discussions. 1.Motivations InthissectionIshalldescribesomeconcreteexampleswhichmotivatethestudy ofhypocoercivity.Allofthemcomefrommathematicalphysics,andnoneofthem isacademic.Ofcoursethelistisfarfromexhaustive. 2