Relative entropy methods for nonlinear diffusion models

Transcription

1 Relative entropy methos for nonlinear iffusion moels p. 1/72 Relative entropy methos for nonlinear iffusion moels Jean Dolbeault CEREMADE CNRS & Université Paris-Dauphine olbeaul EMERGING TOPICS IN DYNAMICAL SYSTEMS AND PARTIAL DIFFERENTIAL EQUATIONS, DSPDES 10, BARCELONA, SPAIN, (MAY 31 JUNE 4, 2010) olbeaul/preprints/

2 Relative entropy methos for nonlinear iffusion moels Outline p. 2/72 Outline Introuction to the notion of entropy A particularly simple case: the heat equation A case with homogeneity: fast iffusion (an porous meia) equations Sharp rates of ecay of solutions to the nonlinear fast iffusion equation via functional inequalities More about homogeneity an the entropy entropy prouction approach Algebraic rates vs. exponential ecay The Bakry-Emery metho revisite The graient flow interpretation From kinetic to iffusive moels Diffusion limit Hypocoercivity Other applications, other moels

3 Relative entropy methos for nonlinear iffusion moels Definition p. 3/72 About entropy in physics Entropy has been introuce as a state function in thermoynamics by R. Clausius in 1865, in the framework of the secon law of thermoynamics, in orer to interpret the the results of S. Carnot A statistical physics approach: Boltzmann s formula (1877) efines the entropy of a systems in terms of a counting of the micro-states of a physical system Boltzmann s equation: t f + v x f = Q(f, f) escribes the evolution of a gas of particles having binary collisions at the kinetic level f is a time epenent istribution function (probability ensity) efine on the phase space R R, thus a function of time t, position x an velocity v. The entropy H[f] := R R f log f xv measures the irreversibility: H-Theorem (1872) t H[f] = Q(f, f) log f xv 0 R R Other notions of entropy: Shannon entropy in information theory, entropy in probability theory (with reference to an arbitrary measure) Other approaches: Carathéoory (1908), Lieb-Yngvason (1997)

4 Relative entropy methos for nonlinear iffusion moels Definition p. 4/72 About entropy in partial ifferential equations In kinetic theory, entropy is one of the few a priori estimates available: it has been use for proucing existence results [DiPerna, Lions], compactness results with application to hyroynamic limits [Baros, Golse, Levermore, Saint-Raymon], convergence of numerical schemes, etc. Nash, Lax, DiPerna: regularity for parabolic equations, hyoynamics, compensate-compactness, geometry, etc. Moeling issues: entropy estimates are compatible with other physical estimates. Exponential convergence is an issue in physics (time-scales), for numerics, for multi-scale analysis For the last 10 years, it has motivate a very large number of stuies in the area of nonlinear iffusions, systems of PDEs, in connection with probability, graient flow an mass transportation techniques It can be use to obtain rates of ecay or intermeiate aymptotics, in connection with functional inequalities Entropy (a loose efinition): a special kin of Lyapunov functional that combines well with other a priori estimates an can be use to investigate the large time behaviour

5 Heat equation an entropy 1. Relative entropy methos for nonlinear iffusion moels Heat equation p. 5/72

6 1. Relative entropy methos for nonlinear iffusion moels Heat equation p. 6/72 Consier the heat equation on the eucliean space R u t = u, u t=0 = u 0 As t, we know that u(t, x) G(t, x) := (4 π t) /2 e x 2 4 t. This is easy to quantify in L or in L 2. How to give (sharp) estimates in L 1? Assume that u 0 0, u 0 (1 + x 2 ), u 0 log u 0 L 1 (R ) an consier the entropy S[u] := R u log u x Time-epenent rescaling an Fokker-Planck equation Relative entropy (free energy), entropy entropy prouction (relative Fisher information) an logarithmic Sobolev inequality Csiszár-Kullback inequality an intermeiate asymptotics The Bakry-Emery metho for proving the logarithmic Sobolev inequality

7 The entropy approach (1/2) Time-epenent rescaling: the change of variables u(τ, y) = R v(t, y/r) with t = log R, R = R(τ) = τ changes the heat equation u τ = u into the Fokker-Planck equation: v t = v + (xv), v t=0 = u 0 with stationary solutions v (x) := (2 π) /2 M e x 2 /2 Relative entropy (free energy): choose M = u R 0 x an efine ( ) v Σ[v] := v log x = v log v x + 1 x 2 v x + Const R v R 2 R If v is a solution of the Fokker-Planck equation, then t Σ[v] = I[v] where I[v] = v v R v + x 2 x is the relative Fisher information Observe that exponential ecay hols by the logarithmic Sobolev inequality [Gross 75] Σ[v] 1 2 I[v] 1. Relative entropy methos for nonlinear iffusion moels Heat equation p. 7/72

8 1. Relative entropy methos for nonlinear iffusion moels Heat equation p. 8/72 The entropy approach (2/2) Large time behaviour is controlle by ( v Σ[v(t, )] = v log R v ) x Σ[u 0 ] e 2 t Using the Csiszár-Kullback inequality v(t, ) v 2 L 1 (R ) 4 M Σ[v] 4 M Σ[u 0] e 2 t we get intermeiate asymptotics for the heat equation, namely u(τ, ) u (τ, ) 2 L 1 (R ) 4 M Σ[v] 4 M Σ[u 0] τ with u (τ, y) := R v (log R, y/r), R = R(τ) = τ Remark: The Bakry-Emery metho gives a proof of the logarithmic Sobolev inequality base on the heat equation: t ( ) I[v] 2 Σ[v] 0

9 Sharp rates of ecay of solutions to the nonlinear fast iffusion equation 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 9/72

10 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 10/72 Fast iffusion equations: outline Introuction Fast iffusion equations: entropy methos an Gagliaro-Nirenberg inequalities [el Pino, J.D.] Fast iffusion equations: the finite mass regime Fast iffusion equations: the infinite mass regime Relative entropy methos an linearization the linearization of the functionals approach: [Blanchet, Bonforte, J.D., Grillo, Vázquez] sharp rates: [Bonforte, J.D., Grillo,Vázquez] An improvement base on the center of mass: [Bonforte, J.D., Grillo, Vázquez] An improvement base on the variance: [J.D., Toscani]

11 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 11/72 Some references J.D. an G. Toscani, Fast iffusion equations: matching large time asymptotics by relative entropy methos, Preprint Matteo Bonforte, J.D., Gabriele Grillo, an Juan-Luis Vázquez. Sharp rates of ecay of solutions to the nonlinear fast iffusion equation via functional inequalities, submitte to Proc. Nat. Aca. Sciences A. Blanchet, M. Bonforte, J.D., G. Grillo, an J.-L. Vázquez. Asymptotics of the fast iffusion equation via entropy estimates. Archive for Rational Mechanics an Analysis, 191 (2): , 02, 2009 A. Blanchet, M. Bonforte, J.D., G. Grillo, an J.-L. Vázquez. Hary-Poincaré inequalities an applications to nonlinear iffusions. C. R. Math. Aca. Sci. Paris, 344(7): , 2007 M. Del Pino an J.D., Best constants for Gagliaro-Nirenberg inequalities an applications to nonlinear iffusions. J. Math. Pures Appl. (9), 81 (9): , 2002

12 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 12/72 Fast iffusion equations: entropy methos u t = u m x R, t > 0 Self-similar (Barenblatt) function: U(t) = O(t /(2 (1 m)) ) as t + [Friemann, Kamin, 1980] u(t, ) U(t, ) L = o(t /(2 (1 m)) ) heat equation fast iffusion equation porous meiaequation m extinction in finite time global existence in L 1 Existence theory, critical values of the parameter m

13 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 13/72 Intermeiate asymptotics for fast iffusion & porous meia Some references Generalize entropies an nonlinear iffusions (EDP, uncomplete): [Del Pino, J.D.], [Carrillo, Toscani], [Otto], [Juengel, Markowich, Toscani], [Carrillo, Juengel, Markowich, Toscani, Unterreiter], [Biler, J.D., Esteban], [Markowich, Leerman], [Carrillo, Vázquez], [Corero-Erausquin, Gangbo, Houré], [Corero-Erausquin, Nazaret, Villani], [Agueh, Ghoussoub],... [el Pino, Sáez], [Daskalopulos, Sesum]... Some methos 1) [J.D., el Pino] relate entropy an Gagliaro-Nirenberg inequalities 2) entropy entropy-prouction metho the Bakry-Emery point of view 3) mass transport techniques 4) hypercontractivity for appropriate semi-groups 5) the approach by linearization of the entropy... Fast iffusion equations an Gagliaro-Nirenberg inequalities We follow the same scheme as for the heat equation

14 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 14/72 Time-epenent rescaling, Free energy Time-epenent rescaling: Take u(τ, y) = R (t) v (t, y/r(τ)) where R τ = R(1 m) 1, R(0) = 1, t = log R The function v solves a Fokker-Planck type equation v t = vm + (xv), v τ=0 = u 0 [Ralston, Newman, 1984] Lyapunov functional: Generalize entropy or Free energy Σ[v] := R ( v m m ) 2 x 2 v x Σ 0 Entropy prouction is measure by the Generalize Fisher information Σ[v] = I[v], I[v] := t R v v m v + x 2 x

15 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 15/72 Relative entropy an entropy prouction Stationary solution: choose C such that v L 1 = u L 1 = M > 0 v (x) := ( C + 1 m ) 1/(1 m) 2 m x 2 + Relative entropy: Fix Σ 0 so that Σ[v ] = 0. The entropy can be put in an m-homogeneous form: for m 1, Σ[v] = R ψ ( ) v v v m x with ψ(t) = tm 1 m(t 1) m 1 Entropy entropy prouction inequality Theorem 1. 3, m [ 1, + ), m > 1 2, m 1 I[v] 2 Σ[v] Corollary 2. A solution v with initial ata u 0 L 1 +(R ) such that x 2 u 0 L 1 (R ), u m 0 L 1 (R ) satisfies Σ[v(t, )] Σ[u 0 ] e 2 t

16 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 16/72 An equivalent formulation: Gagliaro-Nirenberg inequalities Σ[v] = ( ) v m R m x 2 v x Σ v R v m v + x 2 x = 1 2 I[v] Rewrite it with p = 1 2m 1, v = w2p, v m = w p+1 as ( ) 2 ( ) 1 2m 1 w 2 x + 2 2m 1 R 1 m w 1+p x + K 0 R 1 < p = 1 2m 1 2 Fast iffusion case: 1 m < 1 ; K < 0 0 < p < 1 Porous meium case: m > 1, K > 0 for some γ, K = K 0 ( R v x = R w 2p x ) γ w = w = v 1/2p is optimal m = m 1 := 1 : Sobolev, m 1: logarithmic Sobolev Theorem 3. [Del Pino, J.D.] Assume that 1 < p 2 (fast iffusion case) an 3 w L 2p (R ) A w θ L 2 (R ) w 1 θ L p+1 (R ) A = ( y(p 1) 2 2π ) θ 2 ( 2y 2y ) 1 ( ) θ 2p Γ(y) Γ(y 2 ), θ = (p 1) p(+2 ( 2)p), y = p+1 p 1

17 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 17/72 Intermeiate asymptotics Σ[v] Σ[u 0 ] e 2τ + Csiszár-Kullback inequalities Uno the change of variables, with u (t, x) = R (t) v (x/r(t)) Theorem 4. [Del Pino, J.D.] Consier a solution of u t = u m with initial ata u 0 L 1 +(R ) such that x 2 u 0 L 1 (R ), u m 0 L 1 (R ) Fast iffusion case: 1 < m < 1 if 3 lim sup t + t 1 (1 m) 2 (1 m) u m u m L 1 < + Porous meium case: 1 < m < 2 lim sup t + t 1+(m 1) 2+(m 1) [u u ] u m 1 L 1 < +

18 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 18/72 Fast iffusion equations: the finite mass regime Can we consier m < m 1? If m 1: porous meium regime or m 1 := 1 m < 1, the ecay of the entropy is governe by Gagliaro-Nirenberg inequalities, an to the limiting case m = 1 correspons the logarithmic Sobolev inequality Displacement convexity hols in the same range of exponents, m (m 1, 1), as for the Gagliaro-Nirenberg inequalities The fast iffusion equation can be seen as the graient flow of the generalize entropy with respect to the Wasserstein istance if m > m 1 := +2 If m c := 2 m < m 1, solutions globally exist in L 1 an the Barenblatt self-similar solution has finite mass

19 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 19/72...the Bakry-Emery metho We follow the same scheme as for the heat equation Consier the generalize Fisher information I[v] := R v Z 2 x with Z := vm v an compute I[v(t, )]+2 I[v(t, )] = 2 (m 1) u m (ivz) 2 x 2 t R + x i, j=1 R u m ( i Z j ) 2 x the Fisher information ecays exponentially: I[v(t, )] I[u 0 ] e 2t lim t I[v(t, )] = 0 an lim t Σ[v(t, )] = 0 ( ) I[v(t, )] 2 Σ[v(t, )] 0 means I[v] 2 Σ[v] t [Carrillo, Toscani], [Juengel, Markowich, Toscani], [Carrillo, Juengel, Markowich, Toscani, Unterreiter], [Carrillo, Vázquez] I[v] 2 Σ[v] hols for any m > m c

20 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 20/72 Fast iffusion: finite mass regime Inequalities... Sobolev logarithmic Sobolev Gagliaro-Nirenberg m v m L 1, x 2 v L 1 Bakry-Emery metho (relative entropy) global existence in L 1... existence of solutions of u t = u m

21 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 21/72 More references: Extensions an relate results Mass transport methos: inequalities / rates [Corero-Erausquin, Gangbo, Houré], [Corero-Erausquin, Nazaret, Villani], [Agueh, Ghoussoub, Kang] General nonlinearities [Biler, J.D., Esteban], [Carrillo-DiFrancesco], [Carrillo-Juengel-Markowich-Toscani-Unterreiter] an graient flows [Joran-Kinerlehrer-Otto], [Ambrosio-Savaré-Gigli], [Otto-Westickenberg] [J.D.-Nazaret-Savaré], etc Non-homogeneous nonlinear iffusion equations [Biler, J.D., Esteban], [Carrillo, DiFrancesco] Extension to systems an connection with Lieb-Thirring inequalities [J.D.-Felmer-Loss-Paturel, 2006], [J.D.-Felmer-Mayorga] Drift-iffusion problems with mean-fiel terms. An example: the Keller-Segel moel [J.D-Perthame, 2004], [Blanchet-J.D-Perthame, 2006], [Biler-Karch-Laurençot-Nazieja, 2006], [Blanchet-Carrillo-Masmoui, 2007], etc... connection with linearize problems [Markowich-Leerman], [Carrillo-Vázquez], [Denzler-McCann], [McCann, Slepčev], [Kim, McCann], [Koch, McCann, Slepčev]

22 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 22/72 Fast iffusion equations: the infinite mass regime Linearization of the entropy If m > m c := 2 m < m 1, solutions globally exist in L 1 (R ) an the Barenblatt self-similar solution has finite mass. For m m c, the Barenblatt self-similar solution has infinite mass Extension to m m c? Work in relative variables! Σ[V D1 V D0 ] = v 0 V D L 1 V D1 V D0 L 1 V D1 V D0 L1 Σ[V D1 V D0 ] < v 0, V D L Bakry-Emery metho (relative entropy) global existence in L 1 Gagliaro-Nirenberg v m L 1, x 2 v L 1 m m m c m 1

23 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 23/72 Entropy methos an linearization: intermeiate asymptotics, vanishing [A. Blanchet, M. Bonforte, J.D., G. Grillo, J.L. Vázquez], [J.D., Toscani] work in relative variables use the properties of the flow write everything as relative quantities (to the Barenblatt profile) compare the functionals (entropy, Fisher information) to their linearize counterparts = Exten the omain of valiity of the metho to the price of a restriction of the set of amissible solutions Two parameter ranges: m c < m < 1 an 0 < m < m c, where m c := 2 m c < m < 1, T = + : intermeiate asymptotics, τ + 0 < m < m c, T < + : vanishing in finite time lim τրt u(τ, y) = 0 Alternative approach by comparison techniques: [Daskalopoulos, Sesum] (without rates)

24 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 24/72 Fast iffusion equation an Barenblatt solutions u τ = (u um 1 ) = 1 m m um (1) with m < 1. We look for positive solutions u(τ, y) for τ 0 an y R, 1, corresponing to nonnegative initial-value ata u 0 L 1 loc (x) In the limit case m = 0, u m /m has to be replace by log u Barenblatt type solutions are given by U D,T (τ, y) := 1 R(τ) ( D + 1 m 2 m m c y R(τ) 2) 1 1 If m > m c := ( 2)/, U D,T with R(τ) := (T + τ) (m mc) escribes the large time asymptotics of the solutions of equation (1) as τ (mass is conserve) If m < m c the parameter T now enotes the extinction time an R(τ) := (T τ) 1 (mc m) + 1 m If m = m c take R(τ) = e τ, U D,T (τ, y) = e τ ( D + e 2τ y 2 /2 ) /2 Two crucial values of m: m := 4 2 < m c := 2 < 1

25 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 25/72 Rescaling A time-epenent change of variables t := 1 m 2 log ( ) R(τ) R(0) an x := 1 2 m m c y R(τ) If m = m c, we take t = τ/ an x = e τ y/ 2 The generalize Barenblatt functions U D,T (τ, y) are transforme into stationary generalize Barenblatt profiles V D (x) V D (x) := ( D + x 2) 1 m 1 x R If u is a solution to (1), the function v(t, x) := R(τ) u(τ, y) solves v t = [v ( v m 1 V m 1 )] D with initial conition v(t = 0, x) = v 0 (x) := R(0) u 0 (y) t > 0, x R (2)

26 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 26/72 Goal We are concerne with the sharp rate of convergence of a solution v of the rescale equation to the generalize Barenblatt profile V D in the whole range m < 1. Convergence is measure in terms of the relative entropy E[v] := 1 m 1 for all m 0, m < 1 Assumptions on the initial atum v 0 [ v m VD m m V m 1 D R (H1) V D0 v 0 V D1 for some D 0 > D 1 > 0 (v V D ) ] x (H2) if 3 an m m, (v 0 V D ) is integrable for a suitable D [D 1, D 0 ] The case m = m = 4 2 will be iscusse later If m > m, we efine D as the unique value in [D 1, D 0 ] such that (v R 0 V D ) x = 0 Our goal is to fin the best possible rate of ecay of E[v] if v solves (2)

27 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 27/72 Sharp rates of convergence Theorem 5. [Bonforte, J.D., Grillo, Vázquez] Uner Assumptions (H1)-(H2), if m < 1 an m m, the entropy ecays accoring to E[v(t, )] C e 2 (1 m) Λ t t 0 The sharp ecay rate Λ is equal to the best constant Λ α, > 0 in the Hary Poincaré inequality of Theorem 16 with α := 1/(m 1) < 0 The constant C > 0 epens only on m,, D 0, D 1, D an E[v 0 ] Notion of sharp rate has to be iscusse Rates of convergence in more stanar norms: L q (x) for q max{1, (1 m)/ [2 (2 m) + (1 m)]}, or C k by interpolation By unoing the time-epenent change of variables, we euce results on the intermeiate asymptotics of (1), i.e. rates of ecay of u(τ, y) U D,T (τ, y) as τ + if m [m c, 1), or as τ T if m (, m c )

28 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 28/72 Strategy of proof Assume that D = 1 an consier µ α := h α x, h α (x) := (1 + x 2 ) α, with α = 1/(m 1) < 0, an L α, := h 1 α iv [h α ] on L 2 (µ α ): f (L R α, f) µ α 1 = f 2 µ R α A first orer expansion of v(t, x) = h α (x) [ 1 + ε f(t, x) h 1 m α (x) ] solves f t + L α, f = 0 Theorem 6. Let 3. For any α (, 0) \ {α }, there is a positive constant Λ α, such that Λ α, R f 2 µ α 1 R f 2 µ α f H 1 (µ α ) uner the aitional conition f µ R α 1 = 0 if α < α 1 4 ( α)2 if α [ +2 2, α ) (α, 0) Λ α, = 4 α 2 if α [ ), α if α (, ) [Denzler, McCann], [Blanchet, Bonforte, J.D., Grillo, Vázquez]

29 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 29/72 Proof: Relative entropy an relative Fisher information an interpolation For m 0, 1, the relative entropy of J. Ralston an W.I. Newmann an the generalize relative Fisher information are given by F[w] := m 1 m [ R w 1 1 m( w m 1 )] V m D x I[w] := 1 R m 1 [ (w m 1 1) V m 1 D ] 2 v x where w = v V D. If v is a solution of (2), then tf[w(t, )] = I[w(t, )] Linearization: f := (w 1) V m 1 D, h 1 (t) := inf R w(t, ), h 2 (t) := sup R w(t, ) an h := max{h 2, 1/h 1 }. We notice that h(t) 1 as t + h m 2 f 2 V 2 m D x 2 R m F[w] h2 m f 2 V 2 m D x R f 2 V D x [1 + X(h)] I[w] + Y (h) f 2 V 2 m D x R R where X an Y are functions such that lim h 1 X(h) = lim h 1 Y (h) = 0 h 2(2 m) 2 /h 1 h 5 2m =: 1 + X(h) [ (h2 /h 1 ) 2(2 m) 1 ] (1 m) [ h 4(2 m) 1 ] =: Y (h)

30 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 30/72 Proof (continue) A new interpolation inequality: for h > 0 small enough F[w] h2 m [1 + X(h)] 2 [ Λ α, m Y (h) ] m I[w] Another interpolation allows to close the system of estimates: for some C, t large enough, 0 h 1 C F 1 m +2 (+1)m Hence we have a nonlinear ifferential inequality t F[w(t, )] 2 Λ α, m Y (h) [ 1 + X(h) ] h 2 m F[w(t, )] A Gronwall lemma (take h = 1 + C F 1 m +2 (+1)m ) then shows that lim sup t e 2Λ α, t F[w(t, )] < +

31 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 31/72 Plots ( = 5) λ 02 = 8 α 4 ( + 2) λ 30 λ 21 λ12 λ03 Spectrum of Lα, Spectrum of (1 m) L 1/(m 1), λ 11 = 6 α 2 ( + 2) ( = 5) ( = 5) 6 λ 20 = 4 α λ 01 = 4 α 2 4 Essential spectrum of Lα, λ cont α, := 1 4 ( + 2 α 2)2 α = m 2 = m 2 = λ 10 = 2 α α = ( + 2) α = Essential spectrum of (1 m) L 1/(m 1), α = m 1 = 1 α = 1 2 α = m 1 = +2 α = +2 2 α = α m c = 2 1 m

32 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 32/72 Remarks, improvements Optimal constants in interpolation inequalities oes not mean optimal asymptotic rates The critical case (m = m, 3): Slow asymptotics [Bonforte, Grillo, Vázquez] If v 0 V D is boune a.e. by a raial L 1 (x) function, then there exists a positive constant C such that E[v(t, )] C t 1/2 for any t 0 Can we improve the rates of convergence by imposing restrictions on the initial ata? [Carrillo, Leerman, Markowich, Toscani (2002)] Poincaré inequalities for linearizations of very fast iffusion equations (raially symmetric solutions) Formal or partial results: [Denzler, McCann (2005)], [McCann, Slepčev (2006)], [Denzler, Koch, McCann (announcement)], Faster convergence? Improve Hary-Poincaré inequality: uner the conitions R f µ α 1 = 0 an R xf µ α 1 = 0 (center of mas), Λ α, R f 2 µ α 1 R f 2 µ α Next? Can we kill other linear moes?

33 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 33/72 [Bonforte, J.D., Grillo, Vázquez] Assume that m (m 1, 1), 3. Uner Assumption (H1), if v is a solution of (2) with initial atum v 0 such that R xv 0 x = 0 an if D is chosen so that R (v 0 V D ) x = 0, then E[v(t, )] C e γ(m) t t 0 with γ(m) = (1 m) Λ 1/(m 1), γ(m) (= 5) 4 m2 = m2 = m1 = 1 m1 = +2 mc = 2 0 m 1

34 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 34/72 Higher orer matching asymptotics For some m (m c, 1) with m c := ( 2)/, we consier on R the fast iffusion equation u τ + (u u m 1) = 0 The strategy is easy to unerstan using a time-epenent rescaling an the relative entropy formalism. Define the function v such that u(τ, y + x 0 ) = R v(t, x), R = R(τ), t = 1 2 log R, x = y R Then v has to be a solution of v [ t + v ( )] σ 2 (m mc) v m 1 2 x with (as long as we make no assumption on R) 2 σ 2 (m m c) = R 1 (1 m) R τ = 0 t > 0, x R

35 Refine relative entropy Consier the family of the Barenblatt profiles B σ (x) := σ 2 ( CM + 1 σ x 2) 1 m 1 x R (3) Note that σ is a function of t: as long as σ t 0, the Barenblatt profile B σ is not a solution but we may still consier the relative entropy F σ [v] := 1 m 1 [ v m Bσ m m Bσ m 1 R (v B σ ) ] x Let us briefly sketch the strategy of our metho before giving all etails The time erivative of this relative entropy is t F σ(t)[v(t, )] = σ ( ) t σ F σ[v] + m σ=σ(t) m 1 }{{} choose it =0 Minimize F σ [v] w.r.t. σ R x 2 B σ x = R x 2 v x R ( v m 1 B m 1 σ(t) ) v t x (4) 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 35/72

36 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 36/72 Secon step: the entropy / entropy prouction estimate Accoring to the efinition of B σ, we know that 2 x = σ 2 (m m c) B m 1 σ Using the new change of variables, we know that t F σ(t)[v(t, )] = m σ(t) 1 m 2 (m m c) R v [ ] v m 1 B m 1 2 σ(t) x Let w := v/b σ an observe that the relative entropy can be written as F σ [v] = m 1 m R [ w 1 1 m( w m 1 )] B m σ x (Repeating) efine the relative Fisher information by so that I σ [v] := R 1 m 1 [ ] (w m 1 1) Bσ m 1 2 Bσ w x t F σ(t)[v(t, )] = m (1 m) σ(t) I σ(t) [v(t, )] t > 0 When linearizing, one more moe is kille an σ(t) scales out

37 2. Relative entropy methos for nonlinear iffusion moels Fast iffusion p. 37/72 Improve rates of convergence Theorem 7. Let m ( m 1, 1), 2, v 0 L 1 +(R ) such that v m 0, y 2 v 0 L 1 (R ) where E[v(t, )] C e 2 γ(m) t t 0 (( 2) m ( 4)) 2 4 (1 m) if m ( m 1, m 2 ] γ(m) = 4 ( + 2) m 4 if m [ m 2, m 2 ] 4 if m [m 2, 1) γ(m) ( = 5) 4 m2 = m2 = m1 = 1 Case 3 m1 = +2 Case 2 Case 1 mc = 2 0 m 1 [Denzler, Koch, McCann], in progress

38 More about homogeneity 3. Relative entropy methos for nonlinear iffusion moels Homogeneity p. 38/72

39 3. Relative entropy methos for nonlinear iffusion moels Homogeneity p. 39/72 Algebraic rates vs. exponential ecay [J. Carrillo, J.D., I. Gentil, A. Jüngel] Consier the one imensional porous meium/fast iffusion equation u t = (um ) xx, x S 1, t > 0 with u(, t = 0) = u 0 0 The metho also applies to the thin film equation u t = (u m u xxx ) x the Derria-Lebowitz-Speer-Spohn (DLSS) equation u t = (u (log u) xx ) xx Some references: [Cáceres, Carrillo, Toscani], [Gualani, Jüngel, Toscani], [Jüngel, Matthes], [Laugesen] More entropies? p (0, + ), q R, v H 1 +(S 1 ), µ p [v] := ( S 1 v 1/p x ) p [ 1 Σ p,q [v] := p q (p q 1) v q x (µ S 1 p [v]) ] q if p q 1 an q 0 Σ 1/q,q [v] := ( ) v q log R v q x if p q = 1 an q 0 S 1 Σ p,0 [v] := 1 p ( log S 1 S 1 v q x v µ p [v] ) x if q = 0

40 3. Relative entropy methos for nonlinear iffusion moels Homogeneity p. 40/72 Functional inequalities Σ p,q [v] is non-negative by convexity of u up q 1 p q (u 1) p q (p q 1) Proposition 8. Global functional inequalities: For all p (0, + ) an q (0, 2), there exists a positive constant κ p,q such that, for any v H 1 +(S 1 ), Σ p,q [v] 2/q 1 κ p,q S 1 v 2 x Small entropies regime: For any p > 0, q R an ε 0 > 0, there exists a positive constant C such that, for any ε (0, ε 0 ], if v H+(S 1 1 ) is such that Σ p,q [v] ε an µ p [v] = 1 Σ p,q [v] 1 + C ε 8 p 2 π 2 v 2 x S 1

41 3. Relative entropy methos for nonlinear iffusion moels Homogeneity p. 41/72 Application to porous meia: convergence rates u t = (um ) xx x S 1, t > 0 With v := u p, p := m+k 2, q := k+1 p = 2 k+1 m+k, let E[u] := Σ p,q[v] ( 1 k (k+1) S u k+1 ū k+1) x if k R \ { 1, 0} 1 E[u] = u log ( u S ū) x if k = 0 1 log ( u S ū) x if k = 1 1 Proposition 9. Let m (0, + ), k R \ { m}, q = 2 (k + 1)/(m + k), p = (m + k)/2 an u be a smooth positive solution i) Short-time Algebraic Decay: If m > 1 an k > 1, then [ E[u(, t)] E[u 0 ] (2 q)/q + 2 q λκ p,q t q ] q/(2 q) ii) Asymptotically Exponential Decay: If m > 0 an m + k > 0, there exists C > 0 an t 1 > 0 such that for t t 1, E[u(, t)] E[u(, t 1 )] exp ( 8 p2 π 2 λū p(2 q) (t t 1 ) 1 + C E[u(, t 1 )] )

42 3. Relative entropy methos for nonlinear iffusion moels Homogeneity p. 42/72 k m

43 3. Relative entropy methos for nonlinear iffusion moels Homogeneity p. 43/72 The Bakry-Emery metho revisite [J.D., B. Nazaret, G. Savaré] Consier a omain Ω R, γ = g x, g = e F an a generalize Ornstein-Uhlenbeck operator: g v := v DF Dv v t = g v x Ω, t R + v n = 0 x Ω, t R + With s := v p/2 an α := (2 p)/p, p (1, 2] E p (t) := 1 [ ] v p 1 p (v 1) γ p 1 Ω I p (t) := 4 p Ω Ds 2 γ K p (t) := Ω gs 2 γ + α Ω gs Ds 2 s γ A simple computation shows that t E p(t) = I p (t) an t I p(t) = 8 p K p(t)

44 3. Relative entropy methos for nonlinear iffusion moels Homogeneity p. 44/72 Ω K p = An extension of the criterion of Bakry-Emery Using the commutation relation [D, g ] s = D 2 F Ds, we get ( g s) 2 γ = Ω Ω g s 2 γ+4 α Ω D 2 s 2 γ+ D 2 F Ds Ds γ ijs 2 i s n j g H 1 Ω i,j=1 Ω }{{} 0 if Ω is convex g s Dz 2 γ (1 α) Ω with V (x) := D 2 s 2 γ+ V Ds 2 γ Ω ( inf D 2 F(x) ξ, ξ ) ξ S 1 Theorem 10. Let F C 2 (Ω), γ = e F L 1 (Ω), an Ω be a convex omain in R. If λ 1 (p) := inf R Ω 2 p 1 «p Dw 2 +V w 2 R Ω w 2 γ γ is positive, then I p (t) I p (0) e 2 λ 1(p) t E p (t) E p (0) e 2 λ 1(p) t

45 3. Relative entropy methos for nonlinear iffusion moels Homogeneity p. 45/72 Generalize entropies Consier the weighte porous meia equation v t = g v m γ is a probability measure, p (1, 2) 1 [ ] E m,p (t) := v m+p 1 1 γ m + p 2 Ω I m,p (t) := c(m, p) Ds 2 γ K m,p (t) := Ω Ω s β(m 1) g s 2 γ + α Ω s β(m 1) g s Ds 2 s γ with v =: s β, β := 1 p/2+m 1, α := 2 p p+2(m 1) an c(m, p) = 4 m (m+p 1) (2m+p 2) 2

46 3. Relative entropy methos for nonlinear iffusion moels Homogeneity p. 46/72 Aapting the Bakry-Emery metho... Written in terms of s = v 1/β, the evolution is governe by A computation shows that ] 1 m s t = s [ β(m 1) g s + α Ds 2 s 1 m t E m,p(t) t I m,p(t) := I m,p (t) := 2 c(m, p) K m,p (t) Exactly as in the linear case, efine for any θ (0, 1) λ 1 (m, θ) := inf w H 1 (Ω,γ)\{0} Ω ((1 θ) Dw 2 + V w 2 ) Ω w 2 γ γ

47 3. Relative entropy methos for nonlinear iffusion moels Homogeneity p. 47/72 The non-local conition Assume that for some θ (0, 1), λ 1 (m, θ) > 0. Amissible parameters m an p correspon to (m, p) E θ, 1 < m < p + 1, where the set E θ is a portion of an ellipse (grey area) p E 1 E θ 2 1 p=m m

48 3. Relative entropy methos for nonlinear iffusion moels Homogeneity p. 48/72 Results for the porous meia equation Lemma 11. With the above notations, if Ω is convex an (m, p) E θ are amissible, then I 4 3 m,p 1 3 [ [ ] 4 3q ]4 4 c(m, p) 3 K 1 3(2 q) 3 (m + p 2) E m,p + 1 K m,p Theorem 12. Uner the above conitions there exists a positive constant κ which epens on E m,p (0) such that any smooth solution u of the porous meia equation satisfies, for any t > 0 I m,p (t) E m,p (t) I m,p (0) [ κ 3 3 Im,p (0) t] 2 κ 3 [ I m,p (0) ] 8 3 [ 1 + κ 3 3 I m,p (0)t ] 2

49 3. Relative entropy methos for nonlinear iffusion moels Homogeneity p. 49/72 The graient flow interpretation [J.D., B. Nazaret, G. Savaré] As in the Bakry-Emery approach (linear case) let p (1, 2] an efine E p [v] := 1 p 1 R [ ] v p 1 p (v 1) γ, E 1 [v] := R [ ] v log v (v 1) with Ω = R an γ(x) = (2 π) /2 e x 2 /2 x (gaussian measure). Along the flow associate to the Ornstein-Uhlenbeck equation v t = v x v we have foun that E p [v(t, )] E p [v 0 ] e 2 t, which amounts to the generalize Poincaré inequality [Beckner] E p [v] p 2 R v p 2 v 2 γ or R f 2 γ ( R f q γ ) 2/q q 2 R f 2 γ (take q = 2/p, v p = f 2, q [1, 2)) Why o we have such a choice in the linear case? γ

50 3. Relative entropy methos for nonlinear iffusion moels Homogeneity p. 50/72 Wasserstein istances p > 1, µ 0 an µ 1 probability measures on R Transport plans between µ 0 an µ 1 : Γ(µ 0, µ 1 ) is the set of probability measures on R R having µ 0 an µ 1 as marginals. Wasserstein istance between µ 0 ans µ 1 W 2 (µ 0, µ 1 ) = inf { } x y 2 Σ(x, y) : Σ Γ(µ 0, µ 1 ) R R The Benamou-Brenier characterization (2000) { 1 } W 2 (µ 0, µ 1 ) = inf v t 2 ρ t xt : (ρ t,v t ) t [0,1] amissible 0 R where amissible paths (ρ t,v t ) t [0,1] are such that t ρ t + (ρ t v t ) = 0, ρ 0 = µ 0, ρ 1 = µ 1

51 3. Relative entropy methos for nonlinear iffusion moels Homogeneity p. 51/72 Graient flows [Joran, Kinerlehrer, Otto 98] : Formal Riemannian structure on P(R ): the McCann interpolant is a geoesic. For an integral functional such as E 1 [ρ] := F(ρ(x)) x R the graient flow of E 1 w.r.t. W = W 1 is ρ t = [ρ (F (ρ)) ] [Ambrosio, Gigli, Savaré 05] : Rigorous framework for JKO s calculus in the framework of length spaces (base on the optimal transportation) [Otto, Westickenberg 05] : Use the Brenier-Benamou formulation to prove W 2 (µ t 0, µ t 1) W 2 (µ 0, µ 1 ) along the heat flow on a compact Riemannian manifol

52 3. Relative entropy methos for nonlinear iffusion moels Homogeneity p. 52/72 A generalization of the Benamou-Brenier approach Given a function h on R +, efine the amissible paths by { t ρ t + (h(ρ t )v t ) = 0 ρ 0 = µ 0, ρ 1 = µ 1 an consier the istance W 2 p (µ 0, µ 1 ) = inf { 1 0 } v t 2 h p (ρ t ) xt : (ρ t,v t ) t [0,1] amissible R h p (ρ) = ρ 2 p, 1 p 2 p = 1 : Wasserstein case p = 2 : homogeneous Sobolev istance on Ẇ 1,2 { µ 1 µ 0 Ẇ 1,2 = sup ξ (µ 1 µ 0 ) : ξ Cc(R 1 ), R } ξ 2 1 R

53 3. Relative entropy methos for nonlinear iffusion moels Homogeneity p. 53/72 The heat equation as a graient flow w.r.t. W p Denote by S t the semi-group associate to the heat equation. Let p < +2 an consier the generalize entropy functional E p [µ] = 1 p (p 1) R ρ p (x) x if µ = ρ x Theorem 13. If µ P(R ), E p [µ] < +, then E p [S t µ] < + for all t > 0 an 1 2 t W 2 p (S t µ, σ) + E p [S t µ] E p [σ] Beckner inequalities w.r.t. gaussian weight The heat equation can be seen as the graient flow of E p w.r.t. W p

54 From kinetic to iffusive moels 4. Relative entropy methos for nonlinear iffusion moels Diffusion limits, rates p. 54/72

55 4. Relative entropy methos for nonlinear iffusion moels Diffusion limits, rates p. 55/72 We consier a istribution function f = f(t, x, v) solving a non-homogeneous kinetic equation t f + v x f x V v f = Q(f) Generalize entropies i.e. Free energies / entropies / energy uner Casimir constraints are useful to characterize special stationary states un prove their nonlinear stability: [Guo], [Rein], [Schaeffer], etc. Generalize entropies allow to prove existence, characterize large time attractors, take singular limits or quantify the rate of convergence towars an equilibrium [J.D., Markowich, Ölz, Schmeiser]: Diffusion limit of a time-relaxation equation which has polytropic states as stationary solutions [J.D., Mouhot, Schmeiser]: a L 2 hypocoercivity theory... a iffusion limit relate to fast iffusion equations

56 4. Relative entropy methos for nonlinear iffusion moels Diffusion limits, rates p. 56/72 BGK moels BGK moel of gas ynamics ) ρ(x, t) v u(x, t) 2 t f + v x f = exp ( f (2 π T) n/2 2 T(x, t) where ρ(x, t) (position ensity), u(x, t) (local mean velocity) an T(x, t) (temperature) are chosen such that they equal the corresponing quantities associate to f [Perthame, Pulvirenti]: Weighte L bouns an uniqueness for the Boltzmann BGK moel, 1993 Linear BGK moel in semiconuctor physics ρ(x, t) t f + v x f x V v f = exp ( 12 ) (2π) n/2 v 2 f where ρ(x, t) = f(t, x, v) v is the position ensity of f R [Poupau]: Mathematical theory of kinetic equations for transport moelling in semiconuctors, 1994 For applications in astrophysics, we are intereste in collision kernels (BGK type / time-relaxation) such that stationary states are polytropic equilibria ( ) k 1 f(x, v) = 2 v 2 + V (x) µ +

57 4. Relative entropy methos for nonlinear iffusion moels Diffusion limits, rates p. 57/72 Diffusion limit for a time-relaxation moel for polytropes ε 2 t f ε + ε v x f ε ε x V (x) v f ε = G f ε f ε with Gibbs equilibrium G f := γ f ε (x, v, t = 0) = f I (x, v), x, v R 3 ( v 2 ) 2 + V (x) µ ρ f (x, t) The Fermi energy µ ρf (x, t) is implicitly efine by ( ) v V (x) µ ρ f (x, t) v = f(x, v, t) v =: ρ f (x, t) R 3 R 3 γ f ε (x, v, t)... phase space particle ensity V (x)... potential ε... mean free path Formal expansions (generalize Smoluchowski equation): [Ben Aballah, J.D.], [Chavanis, Laurençot, Lemou], [Chavanis et al.], [Degon, Ringhofer] Astrophysics: [Binney, Tremaine], [Guo, Rein], [Chavanis et al.] Fermi-Dirac statistics in semiconuctors moels: [Gouon, Poupau]

58 4. Relative entropy methos for nonlinear iffusion moels Diffusion limits, rates p. 58/72 Fast iffusion an porous meia as a iffusion limit Theorem 14. Uner assumptions on V an the initial ata f I, for any ε > 0, there is a unique weak solution f ε C(0, ; L 1 L p (R 6 )) for all p < which converges to ( ) ( ) 1 1 f 0 (x, v, t) = γ 2 v 2 µ(ρ(x, t)) with 2 v 2 µ(ρ) v = ρ R γ as ε 0, where ρ is a solution of the nonlinear iffusion equation t ρ = x ( x ν(ρ) + ρ x V (x)), ν(ρ) = with initial ata ρ(x, 0) = ρ I (x) := R 3 f I (x, v) v ρ 0 s µ (s) s Fast iffusion case: γ(e) := D E+ k, D > 0 an k > 5/2, ν(ρ) = ρ k 5/2 k 3/2 Linear case: γ(e) := D exp( E), D > 0, ν(ρ) = ρ Porous meium case: γ(e) = D ( E) k +, D > 0 an k > 0, ν(ρ) = ρ k+5/2 k+3/ ,5 8 gamma(e) gamma(e) 1,5 gamma(e) , E E E

59 4. Relative entropy methos for nonlinear iffusion moels Diffusion limits, rates p. 59/72 The relative entropy or free energy functional: with β(s) = 0 s γ 1 (σ) σ convex (γ monotone ecreasing), we get F[f] := R 2 R 2 [ f ( 1 2 v 2 + V ) ] + β(f) xv is such that F[f(t,, )] := t R 2 R 2 ( )( ) G f f γ 1 (G f ) γ 1 (f) xv 0 If f = G ρ is a local Gibbs state, we can efine a reuce free energy by F[G ρ ] = G[ρ] with G ρ (x, v) := γ ( 1 2 v 2 + µ(ρ) ) : G[ρ] = R 2 [h(ρ) + V ρ] x with Polytropes: h(ρ) = 1 m 1 ρm ρ h (ρ) = ν (ρ)

60 4. Relative entropy methos for nonlinear iffusion moels Diffusion limits, rates p. 60/72 Hypocoercivity The goal is to unerstan the rate of relaxation of the solutions of a kinetic equation t f + v x f x V v f = Lf towars a global equilibrium when the collision term acts only on the velocity space. Here f = f(t, x, v) is the istribution function. It can be seen as a probability istribution on the phase space, where x is the position an v the velocity. However, since we are in a linear framework, the fact that f has a constant sign plays no role. A key feature of our approach [J.D., Mouhot, Schmeiser] is that it istinguishes the mechanisms of relaxation at microscopic level (convergence towars a local equilibrium, in velocity space) an macroscopic level (convergence of the spatial ensity to a steay state), where the rate is given by a spectral gap which has to o with the unerlying iffusion equation for the spatial ensity

61 4. Relative entropy methos for nonlinear iffusion moels Diffusion limits, rates p. 61/72 A very brief review of the literature Non constructive ecay results: [Ukai (1974)] [Desvillettes (1990)] Explicit t -ecay, no spectral gap: [Desvillettes, Villani ( )], [Fellner, Miljanovic, Neumann, Schmeiser (2004)], [Cáceres, Carrillo, Gouon (2003)] hypoelliptic theory: [Hérau, Nier (2004)]: spectral analysis of the Vlasov-Fokker-Planck equation [Hérau (2006)]: linear Boltzmann relaxation operator [Prava-Starov], [Hérau, Prava-Starov] Hypoelliptic theory vs. hypocoercivity (Gallay) approach an generalize entropies: [Mouhot, Neumann (2006)], [Villani (2007, 2008)] Other relate approaches: non-linear Boltzmann an Lanau equations: micro-macro ecomposition: [Guo] hyroynamic limits (flui-kinetic ecomposition): [Yu]

62 4. Relative entropy methos for nonlinear iffusion moels Diffusion limits, rates p. 62/72 A toy problem u t = (L T) u, L = ( ), T = ( 0 k k 0 ), k 2 Λ > 0 Nonmonotone ecay, reminiscent of [Filbet, Mouhot, Pareschi (2006)] H-theorem: t u 2 = 2 u 2 2 macroscopic limit: u 1 t = k 2 u 1 generalize entropy: H(u) = u 2 ε k 1+k 2 u 1 u 2 (1 ε) u 2 H(u) (1 ε) u 2 H t = (2 ε k2 1 + k 2 )u 22 ε k2 1 + k 2 u2 1 + ε k 1 + k 2 u 1 u 2 (2 ε) u 2 2 ελ 1 + Λ u2 1 + ε 2 u 1u 2

63 4. Relative entropy methos for nonlinear iffusion moels Diffusion limits, rates p. 63/72 Plots for the toy problem u1 2 1 u u1 2 u H

64 4. Relative entropy methos for nonlinear iffusion moels Diffusion limits, rates p. 64/72... compare to plots for the Boltzmann equation Figure 1: [Filbet, Mouhot, Pareschi (2006)]

65 4. Relative entropy methos for nonlinear iffusion moels Diffusion limits, rates p. 65/72 The kinetic equation t f + T f = L f, f = f(t, x, v), t > 0, x R, v R (5) L is a linear collision operator V is a given external potential on R, 1 T := v x x V v is a transport operator There exists a scalar prouct,, such that L is symmetric an T is antisymmetric t f F 2 = 2 L f 2... seems to imply that the ecay stops when f N(L) but we expect f F as t since F generates N(L) N(T) Hypocoercivity: prove an H-theorem for a generalize entropy H(f) := 1 2 f 2 + ε A f, f

66 4. Relative entropy methos for nonlinear iffusion moels Diffusion limits, rates p. 66/72 Examples, conventions L is a linear relaxation operator L L f = Π f f, Π f := ρ F(x, v) ρ F ρ = ρ f := f v R Maxwellian case: F(x, v) := M(v) e V (x) with M(v) := (2π) /2 e v 2 /2 = Πf = ρ f M(v) Linearize fast iffusion case: F(x, v) := ω ( 1 2 v 2 + V (x) ) (k+1) L is a Fokker-Planck operator: L f = v f + (v f) L is a linear scattering operator (incluing the case of non-elastic collisions) Conventions F is a positive probability istribution Measure: µ(x, v) = F(x, v) 1 xv on R R (x, v) Scalar prouct an norm f, g = R R f g µ an f 2 = f, f

67 4. Relative entropy methos for nonlinear iffusion moels Diffusion limits, rates p. 67/72 Maxwellian case: Assumptions We assume that F(x, v) := M(v) e V (x) with M(v) := (2π) /2 e v 2 /2 where V satisfies the following assumptions (H1) Regularity: V W 2, loc (R ) (H2) Normalization: R e V x = 1 (H3) Spectral gap conition: there exists a positive constant Λ such that R u 2 e V x Λ R x u 2 e V x for any u H 1 (e V x) such that R u e V x = 0 (H4) Pointwise conition 1: there exists c 0 > 0 an θ (0, 1) such that V θ 2 xv (x) 2 + c 0 x R (H5) Pointwise conition 2: there exists c 1 > 0 such that 2 xv (x) c 1 (1 + x V (x) ) x R (H6) Growth conition: R x V 2 e V x <

68 4. Relative entropy methos for nonlinear iffusion moels Diffusion limits, rates p. 68/72 Maxwellian case Theorem 15. If t f + T f = L f, for ε > 0, small enough, there exists an explicit, positive constant λ = λ(ε) such that f(t) F (1 + ε) f 0 F e λ t t 0 The operator L has no regularization property: hypo-coercivity funamentally iffers from hypo-ellipticity Coercivity ue to L is only on velocity variables t f(t) F 2 = (1 Π)f 2 = f ρ f M(v) 2 v x R R T an L o not commute: coercivity in v is transferre to the x variable. In the iffusion limit, ρ solves a Fokker-Planck equation t ρ = ρ + (ρ V ) t > 0, x R The goal of the hypo-coercivity theory is to quantify the interaction of T an L an buil a norm which controls an ecays exponentially Fin a norm which is equivalent to L 2 (µ), for which we have coercivity, base on an operator A The operator A is etermine by the iffusion limit

69 4. Relative entropy methos for nonlinear iffusion moels Diffusion limits, rates p. 69/72 The linearize fast iffusion case Consier a solution of t f + T f = L f where L f = Π f f, Π f := ρ ρ F F F(x, v) := ω ( ) (k+1) 1 2 v 2 + V (x), V (x) = ( 1 + x 2) β where ω is a normalization constant chosen such that R R F xv = 1 an ρ F = ω 0 V /2 k 1 for some ω 0 > 0 Theorem 16. Let 1, k > / There exists a constant β 0 > 1 such that, for any β (min{1, ( 4)/(2k 2)}, β 0 ), there are two positive, explicit constants C an λ for which the solution satisfies: t 0, f(t) F 2 C f 0 F 2 e λt The Poincaré inequality is replace by the Hary-Poincaré inequality associate to the fast iffusion equation The nonlinear case can be reuce to the linear case [Schmeiser, work in progress]

70 Other applications, other moels 5. Relative entropy methos for nonlinear iffusion moels Other applications p. 70/72

71 5. Relative entropy methos for nonlinear iffusion moels Other applications p. 71/72 An uncomplete list Kinetic theory! Molecular motors or moels of Stokes rift: existence, rate of convergence, homogenization, large-time asymptotics: [Kinerlehrer et al.], [J.D., Kowalczyk], [Blanchet, J.D., Kowalczyk], [Perthame, Souganiis] Keller-Segel moels an moels of aggregation in chemotaxis: existence vs. blow-up, critical mass, large time asymptotics, etc. [Blanchet, J.D., Fernánez, Escobeo], [Campos, work in progress] Kinetic an thermoynamical moels for gravitating systems Systems of reaction-iffusion equations in chemistry: [Carrillo, Desvillettes, Fellner] Moels for polymers, fragmentation, etc.

72 p. 72/72 The en... Thank you for your attention!