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TREND TO EQUILIBRIUM FOR DISSIPATIVE EQUATIONS FUNCTIONAL INEQUALITIES AND MASS TRANSPORTATION

15 pages
TREND TO EQUILIBRIUM FOR DISSIPATIVE EQUATIONS, FUNCTIONAL INEQUALITIES AND MASS TRANSPORTATION C. VILLANI In this review paper we shall explain how the trend to equilibrium for some dissipative equations can be studied via certain classes of functional inequalities, with the tool of mass transportation directly or indirectly involved. This short account is mainly intended to give an idea of the subject, a more precise review is given in chapter 9 of [26], or in the research papers [24, 23, 12]. We shall also discuss a few directions of possible future research. 1. A PDE problem Let us consider a set of particles undergoing at the same time diffusion, drift by an external potential force and nonlinear interaction drift. If ?(t, x) stands for the time- dependent density of particles (t ≥ 0, x ? Rn), then a natural evolution model for ? is the partial differential equation (1) ∂?∂t = ∆P (?) +? · (??V ) +? · (??(? ?x W )). Here P (?) stands for the pressure associated with the density ?, V is the external potential, while W is the interaction potential giving rise to the interaction force; it will always be assumed to be symmetric (W (?z) = W (z)). This equation is often called a McKean- Vlasov equation in the particular but most important case P (?) = ?.

  • entropy sense

  • sobolev inequalities

  • equation ∂t?

  • sobolev inequality

  • introduced logarithmic

  • functional inequalities

  • inequality implies

  • fokker-planck equation

  • ?∞


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FUTNRCETNIDONTAOLEIQNUEIQLIUBARLIIUTIMESFOARNDDISMSAIPSSATTIRVEANESQPUOARTTIAOTNISO,NC.VILLANIInthisreviewpaperweshallexplainhowthetrendtoequilibriumforsomedissipativeequationscanbestudiedviacertainclassesoffunctionalinequalities,withthetoolofmasstransportationdirectlyorindirectlyinvolved.Thisshortaccountismainlyintendedtogiveanideaofthesubject,amoreprecisereviewisgiveninchapter9of[26],orintheresearchpapers[24,23,12].Weshallalsodiscussafewdirectionsofpossiblefutureresearch.1.APDEproblemLetusconsiderasetofparticlesundergoingatthesametimediffusion,driftbyanexternalpotentialforceandnonlinearinteractiondrift.Ifρ(t,x)standsforthetime-dependentdensityofparticles(t0,xRn),thenanaturalevolutionmodelforρisthepartialdifferentialequationρ(1)=ΔP(ρ)+∇(ρV)+∇(ρ(ρxW)).tHereP(ρ)standsforthepressureassociatedwiththedensityρ,Vistheexternalpotential,whileWistheinteractionpotentialgivingrisetotheinteractionforce;itwillalwaysbeassumedtobesymmetric(W(z)=W(z)).ThisequationisoftencalledaMcKean-VlasovequationintheparticularbutmostimportantcaseP(ρ)=ρ.ItcanbeconsideredasarelaxedversionofthestandardVlasovequationinkinetictheory(notethat,contrarytothekineticpicture,thereisnovelocityvariablehereinthephasespace).TosuchanequationisassociatedaLyapunovfunctional,or“entropy”,or“freeenergy”,1ZZZ(2)F(ρ)=U(ρ(x))dx+ρ(x)V(x)dx+ρ(x)ρ(y)W(xy)dxdy,2whichcanbedecomposedintothesumofaninternalenergy,apotentialenergyandaninteractionenergy.TherelationbetweenUandPisU′′(ρ)=P(ρ);ofcourseoneshouldimposeU(0)=0.Inparticular,whenP(ρ)=ρonefindsU(ρ)=ρlogρ,sothattheinternalenergycoincideswithBoltzmann’sHfunctional.ToseethatFisaLyapunovfunctionalalong(1),notethatthisequationcanberewrittenas∂ρ=∇ρδF,ρδtwhereδF/δρstandsforthegradientofthefunctionalFwithrespecttotheL2Hilbertstructure.Inviewofthisrewriting,oneseesthat,atleastifwedealwithsmooth,strong2Fδdsolutionsof(1),ZρδtdF(ρ(t))=ρ(t) (t) 0.1