Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck

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1 Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck Jean Dolbeault dolbeaul@ceremade.dauphine.fr CEREMADE CNRS & Université Paris-Dauphine dolbeaul (EN COLLABORATION AVEC C. MOUHOT ET C. SCHMEISER) BCAM, OCTOBER 29, 2009, BILBAO dolbeaul/preprints/ Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.1/28

2 Outline The goal is to understand the rate of relaxation of the solutions of a kinetic equation t f + v x f x V v f = Lf towards a global equilibrium when the collision term acts only on the velocity space. Here f = f(t, x, v) is the distribution function. It can be seen as a probability distribution on the phase space, where x is the position and 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 distinguishes the mechanisms of relaxation at microscopic level (convergence towards a local equilibrium, in velocity space) and macroscopic level (convergence of the spatial density to a steady state), where the rate is given by a spectral gap which has to do with the underlying diffusion equation for the spatial density Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.2/28

3 A very brief review of the literature Non constructive decay results: [Ukai (1974)] [Desvillettes (1990)] Explicit t -decay, no spectral gap: [Desvillettes, Villani ( )], [Fellner, Miljanovic, Neumann, Schmeiser (2004)], [Cáceres, Carrillo, Goudon (2003)] hypoelliptic theory: [Hérau, Nier (2004)]: spectral analysis of the Vlasov-Fokker-Planck equation [Hérau (2006)]: linear Boltzmann relaxation operator Hypoelliptic theory vs. hypocoercivity (Gallay) approach and generalized entropies: [Mouhot, Neumann (2006)], [Villani (2007, 2008)] Other related approaches: non-linear Boltzmann and Landau equations: micro-macro decomposition: [Guo] hydrodynamic limits (fluid-kinetic decomposition): [Yu] Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.3/28

4 A toy problem du dt = (L T) u, L = ( ), T = ( 0 k k 0 ), k 2 Λ > 0 Nonmonotone decay, reminiscent of [Filbet, Mouhot, Pareschi (2006)] H-theorem: d dt u 2 = 2 u 2 2 macroscopic limit: du 1 dt = k 2 u 1 generalized entropy: H(u) = u 2 ε k 1+k 2 u 1 u 2 dh dt = (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 Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.4/28

5 Plots for the toy problem u1 2 1 u u1 2 u H Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.5/28

6 ... compared to plots for the Boltzmann equation Figure 1: [Filbet, Mouhot, Pareschi (2006)] Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.6/28

7 The kinetic equation t f + T f = L f, f = f(t, x, v), t > 0, x R d, v R d (1) L is a linear collision operator V is a given external potential on R d, d 1 T := v x x V v is a transport operator There exists a scalar product,, such that L is symmetric and T is antisymmetric d dt f F 2 = 2 L f 2... seems to imply that the decay 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 generalized entropy H(f) := 1 2 f 2 + ε A f, f Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.7/28

8 Examples L is a linear relaxation operator L L f = Π f f, Π f := ρ F(x, v) ρ F ρ = ρ f := f dv R d Maxwellian case: F(x, v) := M(v) e V (x) with M(v) := (2π) d/2 e v 2 /2 = Πf = ρ f M(v) Linearized fast diffusion 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 (including the case of non-elastic collisions) Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.8/28

9 Some conventions. Cauchy problem F is a positive probability distribution Measure: dµ(x, v) = F(x, v) 1 dxdv on R d R d (x, v) Scalar product and norm f, g = R d R d f g dµ and f 2 = f, f The equation t f + T f = L f with initial condition f(t = 0,, ) = f 0 L 2 (dµ) such that R d R d f 0 dxdv = 1 has a unique global solution (under additional technical assumptions): [Poupaud], [JD, Markowich,Ölz, Schmeiser]. The solution preserves mass R d R d f(t, x, v) dxdv = 1 t 0 Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.9/28

10 Maxwellian case: Assumptions We assume that F(x, v) := M(v) e V (x) with M(v) := (2π) d/2 e v 2 /2 where V satisfies the following assumptions (H1) Regularity: V W 2, loc (Rd ) (H2) Normalization: R d e V dx = 1 (H3) Spectral gap condition: there exists a positive constant Λ such that R d u 2 e V dx Λ R d x u 2 e V dx for any u H 1 (e V dx) such that R d u e V dx = 0 (H4) Pointwise condition 1: there exists c 0 > 0 and θ (0, 1) such that V θ 2 xv (x) 2 + c 0 x R d (H5) Pointwise condition 2: there exists c 1 > 0 such that 2 xv (x) c 1 (1 + x V (x) ) x R d (H6) Growth condition: R d x V 2 e V dx < Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.10/28

11 Maxwellian case: Main result Theorem 1. 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 λt t 0 The operator L has no regularization property: hypo-coercivity fundamentally differs from hypo-ellipticity Coercivity due to L is only on velocity variables d dt f(t) F 2 = (1 Π)f 2 = f ρ f M(v) 2 dv dx R d R d T and L do not commute: coercivity in v is transferred to the x variable. In the diffusion limit, ρ solves a Fokker-Planck equation t ρ = ρ + (ρ V ) t > 0, x R d the goal of the hypo-coercivity theory is to quantify the interaction of T and L and build a norm which controls and decays exponentially Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.11/28

12 The operators. A Lyapunov functional On L 2 (dµ), define bf := Π (v f), af := b (T f), âf := Π ( x f), A := (1+â a Π) 1 â b b f = F v f dv= ρ F R F j f with j f := v f dv ρ d F R d a f = F ) ( x v v f dv + ρ f x V, âf = F x ρ f ρ F R ρ d F A T = (1 + â a Π) 1 â a Define the Lyapunov functional (generalized entropy) H(f) := 1 2 f 2 + ε A f, f Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.12/28

13 ... a Lyapunov functional (continued): positivity, equivalence â a Π f, f = 1 d R d x ( ρf ρ F ) 2 mf dx with m F := R d v 2 F(, v) dv. Let g := A f, u = ρ g /ρ F, j f := R d v f dv (1 + â a Π) g = â bf g 1 d x (m F x u) F ρ F = F ρ F x j f ρ F u 1 d x (m F x u) = x j f A f 2 = u 2 ρ R d F dx and T A f 2 = 1 d R d x u 2 m F dx are such that 2 A f 2 + T A f 2 (1 Π) f 2 As a consequence (1 ε) f 2 2 H(f) (1 + ε) f 2 Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.13/28

14 ... a Lyapunov functional (continued): decay term H(f) := 1 2 f 2 + ε A f, f The operator T is skew-symmetric on L 2 (dµ). If f is a solution, then d H(f F) = D(f F) dt D(f) := f, L f ε A T Π f, f ε A T (1 Π) f, f +ε T A f, f +ε L f, (A+A ) f }{{}}{{} micro macro Lemma 2. For some ε > 0 small enough, there exists an explicit constant λ > 0 such that D(f F) + λh(f F) 0 Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.14/28

15 Preliminary computations (1/2) Maxwellian case: Πf = ρ f M(v) with ρ f := R d f dv Replace f by f F : 0 = R d R d f dxdv = f, F, R d (Π f f) dv = 0 L is a linear relaxation operator: L f = Π f f The other terms: for any c, Lf, f (1 Π) f 2 ε A T (1 Π) f, f = ε A T (1 Π) f, Π f c 2 A T (1 Π) f 2 + ε2 2 c ε T A f, f = ε T A f, Π f c 2 T A f 2 + ε2 2 c ε (A + A ) L f, f ε (1 Π) f 2 + ε A f 2 Π f 2 Π f 2 Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.15/28

16 Reminder: A and T A are bounded operators A T (1 Π) = Π A T (1 Π) and T A = Π T A. With m F := R d v 2 F dv â a Π f, f = 1 d R d x ( ρf ρ F ) 2 mf dx Recall that one ca compute A f := (1 + â a Π) 1 â b f as follows: let g := A f u := ρ g /ρ F By definition of A, â bf = (1 + â a Π) g means x v f dv =: x j f = ρ F u 1 R d x (m F x u) d A f 2 = u 2 ρ R d F dx and T A f 2 = 1 d R d x u 2 m F dx are such that 2 A f 2 + T A f 2 (1 Π) f 2 Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.16/28

17 Preliminary computations (2/2) D(f) (1 c ) 2 2 ε (1 Π) f 2 }{{} micro: 0 ε A T Π f, f }{{} macro: first estimate + c 2 A T (1 Π) f 2 }{{} second estimate... + ε2 c Π f 2...(A T (1 Π)) f = (â a (1 Π)) g with g = (1 + â a Π) 1 f means ρ f = ρ F u 1 d x (m F x u) where u = ρ g /ρ F. Let q F := R d v 1 4 F dv, u ij := 2 u/ x i x j (A T (1 Π)) f 2 = d i, j=1 R d [( ) ] 2 δij +1 3 q F m2 F δ ij d 2 ρ F u ii u jj + 2(1 δ ij) 3 q F u 2 ij dx Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.17/28

18 Maxwellian case With ρ F = e V = 1 d m F = q F, B = â a Π, g = (1 + B) 1 f means ρ f = u e V x ( e V x u ) if u = ρ g ρ F Spectral gap condition: there exists a positive constant Λ such that R d u 2 e V dx Λ R d x u 2 e V dx Since A T Π = (1 + B) 1 B, we get the first estimate" A T Π f, f }{{} macro Λ 1 + Λ Π f 2 Notice that B = â a Π = (T Π) (T Π) so that B f, f = T Π f 2 Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.18/28

19 Second estimate (1/3) We have to bound (H 2 estimate) (A T (1 Π)) f 2 2 d i, j=1 R d u ij 2 e V dx Let u 2 0 := R d u 2 e V dx. Multiply ρ f = u e V x ( e V x u ) by u e V to get u x u 2 0 Π f 2 By expanding the square in x (u e V/2 ) 2, with κ = (1 θ)/(2 (2 + Λ c 0 )), we obtain an improved Poincaré inequality κ W u 2 0 x u 2 0 for any u H 1 (e V dx) such that R d u e V dx = 0 Here: W := x V Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.19/28

20 Second estimate (2/3) Multiply ρ f = u e V x ( e V x u ) by W 2 u with W := x V and integrate by parts W u W x u c 1 ( x u 0 + W x u 0 ) W u 0 κ 8 W 2 u κ Π f 2 Improved Poincaré inequality applied to W u W u e V dx gives R d κ W 2 u 2 0 x (W u) 2 e V dx + 2 κ W u e V dx W 3 u e V dx R d R d R d (...) κ W 2 u W x u c 2 1 ( u W u 2 0) + 4 κ W 4 0 u 2 0 W x u 0 c 5 Π f Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.20/28

21 Second estimate (3/3) Multiply ρ f = u e V x ( e V x u ) by u and integrating by parts, we get 2 xu 2 0 ( W x u 0 + Π f ) 2 xu 0 W u 0 x u 0 Altogether (...) (A T (1 Π)) f 2 c 6 (1 Π) f 2 Summarizing, with λ 1 = 1 c 2 (1 + c 6) 2 ε and λ 2 = Λ ε 1+Λ ε2 c D(f) λ 1 (1 Π) f 2 λ 2 Π f 2 Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.21/28

22 The Vlasov-Fokker-Planck equation t f + T f = L f with L f = v f + v (v f) Under the same assumptions as in the linear BGK model... same result! A list of changes f, L f }{{} micro = v f 2 (1 Π) f 2 by the Poincaré inequality Since Π f = ρ f M(v), where M(v) is the gaussian function, and V (x) F(x, v) = M(v) e A f, L f = ρ A f M(v) (L f) dxdv F = ρ A f (L f) e V dxdv= 0 A f = u F means ρ F u 1 d x (m F x u) = x j f, j f := R d v f dv. Hence j L f = j f gives A Lf, f = A f, f Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.22/28

23 Motivation: nonlinear diffusion as a diffusion limit [ ] ε 2 t f + ε v x f x V (x) v f with Q(f) = γ Local mass conservation determines µ(ρ) ( ) 1 2 v 2 µ(ρ f ) = Q(f) Theorem [Dolbeault, Markowich, Oelz, CS, 2007] ρ f converges as ε 0 to a solution of t ρ = x ( x ν(ρ) + ρ x V ) with ν (ρ) = ρ µ (ρ) f γ(s) = ( s) k +, ν(ρ) = ρ m, 0 < m = m(k) < 5/3 (R 3 case) Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.23/28

24 Linearized fast diffusion case Consider a solution of t f + T f = L f where L f = Π f f, Π f := ρ ρ 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 d R d F dxdv = 1 and ρ F = ω 0 V d/2 k 1 for some ω 0 > 0 Theorem 3. Let d 1, k > d/ There exists a constant β 0 > 1 such that, for any β (min{1, (d 4)/(2k d 2)}, β 0 ), there are two positive, explicit constants C and λ for which the solution satisfies: t 0, f(t) F 2 C f 0 F 2 e λt. Computations are the same as in the Maxwellian case except for the first estimate" and the second estimate" Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.24/28

25 Linearized fast diffusion case: first estimate For p = 0, 1, 2, let w 2 p := ω 0 V p q, where q = k + 1 d/2, w 2 0 := ρ F u 2 i = Now g = (1 + â a Π) 1 f means R d u 2 w 2 i dx ρ f = w 2 0 u 2 2k d x ( w 2 1 x u ) Hardy-Poincaré inequality [Blanchet, Bonforte, J.D., Grillo, Vázquez] u 2 0 Λ x u 2 1 under the condition R d u w 2 0 dx = 0 if β 1. As a consequence A T Π f, f Λ 1 + Λ Π f 2 Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.25/28

26 Linearized fast diffusion case: second estimate Observe that ρ f = w 2 0 u 2 2k d x ( w 2 1 x u ) multiplied by u gives u (q 1) 1 x u 2 1 Π f 2 By the Hardy-Poincaré inequality (condition β < β 0 ) R d V α+1 q 1 β u 2 dx ( R d V α+1 q 1 β u dx ) 2 R d V α+1 q 1 β dx 1 4 (β 0 1) 2 R d V α+1 q x u 2 dx By multiplying ρ f = w 2 0 u 2 2k d x ( w 2 1 x u ) by V α u with α := 1 1/β or by V u we find 2 xu 2 2 C Π f 2 Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.26/28

27 Diffusion limits and hypocoercivity The strategy of the method is to introduce at kinetic level the macroscopic quantities that arise by taking the diffusion limit kinetic equation diffusion equation functional inequality (macroscopic) Vlasov + BGK / Fokker-Planck Fokker-Planck Poincaré (gaussian weight) linearized Vlasov-BGK linearized porous media Hardy-Poincaré nonlinear Vlasov-BGK porous media Gagliardo-Nirenberg from kinetic to diffusive scales: parabolic scaling and diffusion limit heuristics: convergence of the macroscopic part at kinetic level is governed by the functional inequality at macroscopic level interplay between diffusion limits and hypocoercivity is still work in progress as well as the nonlinear case Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.27/28

28 Concluding remarks hypo-coercivity vs. hypoellipticity diffusion limits, a motivation for the fast diffusion case other collision kernels: scattering operators other functional spaces nonlinear kinetic models hydrodynamical models Reference J. Dolbeault, C. Mouhot, C. Schmeiser, Hypocoercivity for kinetic equations with linear relaxation terms, CRAS 347 (2009), pp dolbeaul/conferences/ Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck p.28/28

29 Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model Jean Dolbeault CNRS & Université Paris-Dauphine dolbeaul (A JOINT WORK WITH A. BLANCHET, M. ESCOBEDO AND J. FERNÁNDEZ) BCAM, OCTOBER 30, 2009, BILBAO dolbeaul/preprints/ Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.1/46

30 Outline Introduction: the two-dimensional parabolic-elliptic Keller-Segel model, the M < 8π regime, scalings, etc First digression : functional inequalities, rates, etc Second digression : relative entropies and linearization The asymptotic behaviour of the solutions of the Keller-Segel model References on the general theory of Keller-Segel systems: [ Horstmann ] Disclaimer: many references! Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.2/46

31 Introduction Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.3/46

32 The parabolic-elliptic Keller and Segel system u t = u (u v) x R2, t > 0 v = u x R 2, t > 0 u(, t = 0) = n 0 0 x R 2 We make the choice: v(t, x) = 1 2π R 2 log x y u(t, y) dy and observe that Mass conservation: v(t, x) = 1 2π d dt R 2 R 2 u(t, x) dx = 0 x y u(t, y) dy x y 2 Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.4/46

33 Blow-up M = n R 2 0 dx > 8π and x 2 n R 2 0 dx < : blow-up in finite time a solution u of u = u (u v) t satisfies d dt x 2 u(t, x) dx R 2 = 2x u dx + 1 R 2π 2 u(t, x) u(t, y) dxdy }{{} 2x (y x) x y 2 R 2 R 2 (x y) (y x) x y 2 u(t,x) u(t,y) dx dy = 4 M M2 2π < 0 if M > 8π Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.5/46

34 Existence and free energy M = R 2 n 0 dx 8π: global existence [ Jäger, Luckhaus ], [ JD, Perthame ], [ Blanchet, JD, Perthame ], [ Blanchet, Carrillo, Masmoudi ] If u solves the free energy satisfies u t F[u] := = [u ( (log u) v)] R 2 u log u dx 1 2 R 2 u v dx d F[u(t, )] = u (log u) v 2 dx dt R 2 Log HLS inequality [ Carlen, Loss ]: F is bounded from below if M < 8π Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.6/46

35 The dimension d = 2 In dimension d, the norm L d/2 (R d ) is critical. If d = 2, the mass is critical Scale invariance: if (u, v) is a solution in R 2 of the parabolic-elliptic Keller and Segel system, then is also a solution ( ) λ 2 u(λ 2 t, λx), v(λ 2 t, λx) For M < 8π, the solution vanishes as t, but saying that "diffusion dominates" is not correct: to see this, study "intermediate asymptotics" Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.7/46

36 The existence setting u t = u (u v) x R2, t > 0 v = u x R 2, t > 0 u(, t = 0) = n 0 0 x R 2 Initial conditions n 0 L 1 +(R 2, (1+ x 2 ) dx), n 0 log n 0 L 1 (R 2, dx), M := R 2 n 0 (x) dx < 8 π Global existence and mass conservation: M = R 2 u(x, t) dx for any t 0, see [ Jäger-Luckhaus ], [ Blanchet, JD, Perthame ] v = 1 2π log u Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.8/46

37 Time-dependent rescaling u(x, t) = 1 ( ) x R 2 (t) n R(t), τ(t) and v(x, t) = c ( ) x R(t), τ(t) with R(t) = 1 + 2t and τ(t) = log R(t) n t = n (n ( c x)) x R2, t > 0 c = 1 2π log n x R2, t > 0 n(, t = 0) = n 0 0 x R 2 [ Blanchet, JD, Perthame ] Convergence in self-similar variables lim n(, + t) n t L1 (R 2 ) = 0 and lim c(, + t) c t L2 (R 2 ) = 0 means "intermediate asymptotics" in original variables: ( ) u(x, t) 1 R 2 (t) n x R(t), τ(t) L1 (R 2 ) ց 0 Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.9/46

38 The stationary solution in self-similar variables n = M e c x 2 /2 e R c x 2 /2 dx = c, c = 1 2π log n 2 Radial symmetry [ Naito ] Uniqueness [ Biler, Karch, Laurençot, Nadzieja ] As x +, n is dominated by e (1 ǫ) x 2 /2 for any ǫ (0, 1) [ Blanchet, JD, Perthame ] Bifurcation diagram of n L (R 2 ) as a function of M: lim n M 0 L (R 2 ) = 0 + [ Joseph, Lundgreen ] [ JD, Stańczy ] Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.10/46

39 The free energy in self-similar variables satisfies F[n] := n [ ] t = n (log n x + c) R 2 n log n dx + R x 2 n dx 1 2 R 2 nc dx d F[n(t, )] = n (log n) + x c 2 dx dt R 2 A last remark on 8π and scalings: n λ (x) = λ 2 n(λx) F[n λ ] = F[n]+ n log(λ 2 λ ) dx x 2 n dx+ 1 R 2 R 4π 2 F[n λ ] F[n] = ) (2M M2 4π }{{} >0 if M<8π log λ + λ R 2 R 2 n(x) n(y) log 1 λ dxdy R 2 x 2 n dx Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.11/46

40 Main result Theorem 1. There exists a positive constant M such that, for any initial data n 0 L 2 (n 1 dx) of mass M < M satisfying the above assumptions, there is a unique solution n C 0 (R +, L 1 (R 2 )) L ((τ, ) R 2 ) for any τ > 0 Moreover, there are two positive constants, C and δ, such that R 2 n(t, x) n (x) 2 dx n C e δ t t > 0 As a function of M, δ is such that lim M 0+ δ(m) = 1 The condition M 8 π is necessary and sufficient for the global existence of the solutions, but there are two extra smallness conditions in our proof: Uniform estimate: the method of the trap Spectral gap of a linearised operator L Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.12/46

41 Digression 1: Functional inequalities, rates Some comments on the Fokker-Planck equation n t = n + (xn) Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.13/46

42 L 2 framework Spectral decomposition: use Hermite functions. Spectrum is made of all positive integers and n (x) = M 2 /2 e x (2π) d/2 1 2 d dt is the unique stationary solution (in the kernel). Return to equilibrium: if you are interested only in the rate of return to equilibrium, use spectral gap / Poincaré inequality with gaussian weight n R 2 n n 1 2 n dx = R 2 ( ) n 2 n n dx R 2 n n 1 2 n dx n n 2 n n L 2 n ( n 0 L + n L ) C e t R 2 n Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.14/46

43 L 1 framework d dt Return to equilibrium: use gaussian logarithmic Sobolev inequality ( n n log R 2 n ) dx = R 2 ( n log n and conclude by the Csiszár-Kullback inequality n n 2 L 1 4 M ( n n log R 2 ) 2 ( n n dx 2 n log R 2 n ) dx C e 2t Simpler! If you are only interested in the asymptotic rates, observe that s log s s + 1 s 1 2 n [ Bartier, Blanchet, J.D., Escobedo ] ) dx Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.15/46

44 Digression 2: Relative entropies and linearization: fast diffusions Some comments on the fast diffusion (m < 1) equation (in self-similar variables) n t = nm + (xn) Convergence towards the stationary solution? 1 1 d m < 1: Gagliardo-Nirenberg inequalities [ del Pino, J.D. ] and many others 1 2 d m < 1: entropy - entropy production approach (Bakry-Emery method) [ Carrillo et al. ] Linearization [ Blanchet, Bonforte, J.D., Grillo, Vázquez ] Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.16/46

45 The free energy for fast diffusions With w := u/u, we define the relative entropy F[w] := 1 1 m R d [ (w 1) 1 m (wm 1) ] u m dx the relative Fisher information m I[w] := [( w m 1 1 ) u m 1 ] (m 1) 2 2 w u dx R d d F[w(t)] = I[w(t)] dt Linearized entropy and Fisher information functionals: w = 1 + ε F[g] := 1 2 R d g 2 u 2 m dx and I[g] := m R d g 2 u dx g u m 1 Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.17/46

46 Hardy-Poincaré inequalities and asymptotic behaviour Hardy-Poincaré inequalities: for any m m = (d 4)/(d 2), m (0, 1), for any g D(R d ), F[g] C m,d I[g] under the condition R d g u 2 m Relative entropy Fisher information dx = 0 if m > m C 1 F[g] F[w] C 2 F[g] I[g] β 1 I[w] + β 2 F[g] with g := (w 1) u m 1 Conclusion d dt F[w(t)] = I[w(t)] 1 β 1 I[g]+ β 2 β 1 F[g] 1 β 1 C 2 ( ) 1 β 2 F[w(t)] C m,d Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.18/46

47 Back to the Keller-Segel model Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.19/46

48 Proof of the main result First step: the trap Second step: weighted H 1 estimates Third step: linearization and spectral gap Fourth step: collecting the estimates Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.20/46

49 The parabolic-elliptic Keller and Segel system u t = u (u v) x R2, t > 0 v = u x R 2, t > 0 u(, t = 0) = n 0 0 x R 2 Initial conditions n 0 L 1 +(R 2, (1+ x 2 ) dx), n 0 log n 0 L 1 (R 2, dx), M := R 2 n 0 (x) dx < 8 π Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.21/46

50 First step: the trap Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.22/46

51 Decay Estimates of u(t) in L (R 2 ) Lemma 2. For any M < M 1, there exists C = C(M) such that, for any solution u C 0 (R +, L 1 (R 2 )) L (R + loc R2 ) The method of the trap... prove that u(t) L (R 2 ) C t 1 t > 0 H(ψ(t), M) 0 where ψ(t) := t u(, t) L (R 2 ) where z H(z, M) is a continuous function which is - negative on [0, z 1 ) - positive on (z 1, z 2 ) for some z 1, z 2 such that 0 < z 1 < z 2 < ψ is continuous and ψ(0) = 0 = ψ(t) z 1 z 0 (M) for any t 0 if H(z 0 (M), M) = sup z [z1,z 2 ] H(z, M) 0 Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.23/46

52 H(z 0 (M),M) H(z, M) 0 z 0 (M) z z 0 (M 0 (p)) M 2π M 0(p) 2π H(z, M 0 (p)) The method of the trap amounts to prove that H(z, M) 0 implies that z = ψ(t) is bounded by z 0 (M) as long as H(z 0 (M), M) > 0 Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.24/46

53 Duhamel s formula: u(x, t 0 + t) = t 0 N(x y, t) u(y, t 0 ) dy R 2 R 2 N(x y, t s) [u(y, t 0 + s) v(y, t 0 + s)] dy ds where N(x, t) = 1 4πt e x 2 /(4t). Let κ σ = N/ x i (, 1) Lσ (R 2 ) u(, t 0 + t) L (R 2 ) 1 4πt u(, t 0) L1 (R 2 ) t N [( (, t s) u v ) (, t 0 + s)] x i x i i=1,2 0 i=1,2 t κ σ (t s) (1 1 σ ) L (R 2 ) ( u v ) (, t 0 + s) x i ds L ρ (R 2 ) ds Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.25/46

54 HLS inequality + Hölder and take t 0 = t 2t u(, 2t) L (R 2 ) M 2π 2 κ σ C HLS π M 1 p + 1 r t t 0 [ ] (t s) 1 σ (t + s) p r 2 p ψ(t) 1 r ds with ψ(t) := sup 0 s t 2s u(, 2s) L (R 2 ) and t t 0 (t s) 1 σ 3 2 (t + s) 1 p + 1 r 2 ds = σ 2 σ ψ(t) M 2π +C 0 ( ψ(t) ) θ with C 0 = 2 κ σ C HLS π M 1 p + 1 r σ 2 σ, θ = 2 1 p 1 r Choice: H(z, M) = z C 0 z θ M/(2π) Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.26/46

55 How small is the mass? The exponents σ, ρ, p, q and r are related by 1 σ + 1 ρ = 1, 1 < σ < 2 1 p + 1 q = 1 ρ, p, q > 2 1 r 1 q = 1 2, r > 1 For the choice r = 4/3, q = 4, C HLS = 2 π C 0 = 4κ σ π M 1 p +1 4 σ 2 σ with σ = 4p 3p 4... there exists M 0 (p) such that H(z 0 (M), M) > 0 if and only if M < M 0 (p) and sup p (4,+ ) M 0 (p) = lim p + M 0 (p) < 8π Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.27/46

56 Other norms: interpolation Corollary 3. For any mass M < M 1 and all p [1, ], there exists a positive constant C = C(p, M) with lim M 0+ C(p, M) = 0, such that u(t) L p (R 2 ) C t (1 1 p ) t > 0 Remark 1. The above rates are optimal as can easily be checked using the self-similar solutions (n, c ) Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.28/46

57 Second step: weighted H 1 estimates Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.29/46

58 L p and H 1 estimates in the self-similar variables Consider the solution of n t = n (n ( c x)) x R2, t > 0 c = 1 2π log n x R2, t > 0 n(, t = 0) = n 0 0 x R 2 For any p (1, ] n(t) L p (R 2 ) C 1 t > 0 for some positive constant C 1, and for p > 2 2π c(t) L n(t, y) sup x R 2 x y 1 x y dy }{{} M + sup n(t, y) x R 2 x y 1 x y dy }{{} (2π p 1 p 2) p p 1 n L p (R 2 ) Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.30/46

59 n(t) L p (R 2 ) C 1 and c(t) L (R 2 ) C 2 t > 0 Lemma 4. The constants C 1 and C 2 depend on M and are such that lim C i (M) = 0 i = 1, 2 M 0 + Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.31/46

60 Exponential weights With K = K(x) = e x 2 /2, let us rewrite the equation for n as n t 1 K (K n) = c n + 2n + n2 Proposition 5. For any mass M (0, M 1 ), there is a positive constant C such that n(t) H1 (K) C t > 0 First ingredient [ M. Escobedo and O. Kavian ]: for any q > 2 and ε > 0, there exists a positive constant C(ε, q) such that n 2 K dx ε n 2 K dx + C(ε, q) n 2 L q (R 2 ) R 2 R 2 Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.32/46

61 L 2 (K) estimate Multiply the equation by n K and integrate by parts 1 2 d dt n 2 K dx + n 2 K dx R 2 R 2 = n c n K dx + 2 n 2 K dx + n 3 K dx R 2 R 2 R 2 ε n 2 K dx + C R 2 and so 1 2 d dt n 2 K dx + (1 ε) n 2 K dx C R 2 R } 2 {{} R R 2 n 2 K dx (expand the square in R 2 (nk) 2 K 1 dx 0) Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.33/46

62 H 1 (K) estimate (1/2) Let S(t) be the semi-group generated by K 1 (K ) on L 2 (K) t t n(t, x) = S(t) n 0 (x) S(t s) ( c n)(s) ds+ S(t s) (2n+n 2 )(s) ds 0 0 n(t) H1 (K) S(t) n 0 H1 (K) t 0 S(t s) ( c n)(s) H1 (K) ds+ t 0 S(t s) (2n+n 2 )(s) H1 (K) ds Second ingredient: S(t) h H1 (K) κ (1 + t 1/2 ) h L2 (K) 1 κ ( n(t) H 1 (K) S(t) n 0 H 1 (K)) C 2 t 0 ( ) t t s n(s) L2 (K) ds + (2 + C 1 ) 0 ( t s ) n(s) L2 (K) Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.34/46

63 H 1 (K) estimate (2/2) 1 κ n(t + τ) H 1 (K) Let H(T) = sup t (0,T) t 0 such that 1 2κ = C 3 T 0 (1 + 1 t ) C 1 + C 3 t 0 ( t s ) n(s + τ) H1 (K) ds ( ) t s n(s + τ) H1 (K) ds and choose T > 0 ( ) ( ) T s ds = C 3 T + 2 T 1 ( κ H(T) π + 4 ) T + T C ( 2κ H(T) = H(T) 2 π + 4 ) T + T κ C 1 For any t (0, T) 1 κ n(t+τ) H 1 (K) (1 + 1 t ) C 1 +C 3 H(T) (1 + 1 t ) C 1 +2 ( π + 4 ) T + T Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.35/46

64 Conclusion: bound in H 1 (K) The estimate 1 κ n(t+τ) H 1 (K) (1 + 1 t ) C 1 +C 3 H(T) (1 + 1 t ) C 1 +2 ( π + 4 ) T + T for any t (0, T) gives a bound on n(t + τ) H1 (K) for any τ > 0 Lemma 6. n(t) H 1 (K) {1, C max } T t t > 0 Actually n(t) can be bounded also in H 1 (n 1 ) but further estimates are needed... Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.36/46

65 Third step: linearization and spectral gap Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.37/46

66 A linearized operator Introduce f and g defined by n(x, t) = n (x)(1 + f(x, t)) and c(x, t) = c (x)(1 + g(x, t)) (f, g) is solution of the non-linear problem f t L(t, x, f, g) = 1 n [f n (g c )] x R 2, t > 0 (c g) = f n x R 2, t > 0 where L is the linear operator given by L(t, x, f, g) = 1 n [n (f g c )] The conservation of mass is replaced here by R 2 f n dx = 0 Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.38/46

67 A spectral gap estimate Proposition 7. For any M (0, M 2 ), for any f H 1 (n dx) such that f n dx = 0 = f 2 n dx Λ(M) f 2 n dx R 2 R 2 R 2 for some Λ(M) > 0 and lim M 0+ Λ(M) = 1 Let h = n f = λe x 2 /4+c /2 f λ f 2 n = h 2 + x 2 4 h c 2 h 2 +h h (x c ) 1 2 x c h 2 (integrations by parts) h h x dx = h 2 dx R2 R 2 h h c R 2 dx = 1 2 h 2 ( c R 2 ) dx 1 2 n L (R 2 ) h 2 dx R x c R 2 h 2 dx σ2 1 x 2 σ 2 R 2 4 h2 dx + 1 σ 2 4 σ 2 1 c R 2 2 h 2 dx Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.39/46

68 H 1 (n 1 ) estimate Assume that n 0 /n L 2 (n ) There exists a constant C > 0 such that x > 1 = c + M/(2π) log x C n K = e c behaves like O( x M/(2π) ) as x n t n ( ) 1 n n = ( c c) n + 2n + n 2 Corollary 8. If M < M 2, then any solution n is bounded in for any τ > 0 L (R +, L 2 (n 1 dx)) L ((τ, ), H 1 (n 1 dx)) Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.40/46

69 Fourth step: collecting the estimates Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.41/46

70 Proof of the exponential rate of convergence f t L(t, x, f, g) = 1 n [f n (g c )] x R 2, t > 0 (c g) = f n x R 2, t > 0 Multiply by f n and integrate by parts 1 2 d dt f 2 n dx + f 2 n dx R 2 R 2 = f (g c ) n dx+ f (g c ) f n dx R } 2 R {{}} 2 {{} =I =II Cauchy-Schwarz inequality I = R 2 f (g c ) n dx f L2 (n dx) (g c ) L2 (n dx) Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.42/46

71 first term Hölder s inequality (with q > 2) (g c ) L2 (n dx) M1/2 1/q n 1/q L (R 2 ) (g c ) Lq (R 2 ) HLS inequality (with 1/p = 1/2 + 1/q) (g c ) Lq (R 2 ) 1 2π ( R 2 (f n ) 1 ) q 1 q C HLS dx 2π f n Lp (R 2 ) Hölder s inequality: f n L p (R 2 ) f L 2 (n dx) n 1/2 L q/2 (R 2 ) I = R 2 f (g c ) f n dx C (M) f L 2 (n dx) f L2 (n dx) C (M) := C HLS (2π) 1 M 1/2 1/q n 1/2 L q/2 (R 2 ) n 1/q L (R 2 ) 0 as M 0 Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.43/46

72 second term and conclusion Use g c = c c and the Cauchy-Schwarz inequality f (g c ) f n dx c c L (R 2 ) R 2 }{{} 2 C 2 (M)ց0 f L2 (n dx) f L2 (n dx) Spectral gap estimate Λ(M) }{{} 1 f L 2 (n dx) f L 2 (n dx) With γ(m) := C (M)+2 C 2 (M) ց 0, Λ(M) 1 2 d dt f 2 n dx [1 γ(m)] f 2 n dx R 2 R 2 Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.44/46

73 Uniqueness If n 1 and n 2 are two solutions in C 0 (R +, L 1 (R 2 )) L ((τ, ) R 2 ) for any τ > 0, with f = (n 2 n 1 )/n we also get 1 2 d dt R 2 f 2 n dx [1 γ(m)]λ(m) R 2 f 2 n dx As a consequence, if the initial condition is the same, then n 1 = n 2 Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.45/46

74 Thank you for your attention! Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model p.46/46

75 Large time asymptotics of simple ratchet models Jean Dolbeault (Ceremade, Université Paris-Dauphine) (with A. Blanchet et M. Kowalczyk) dolbeaul BCAM, october 30, 2009, Bilbao

76 Travelling fronts in stochastic Stokes drifts and Brownian ratchets: homogenized functional inequalities and large time behaviour of the solutions Outline Introduction to ratchets models Homogenized functional inequalities Traveling and tilted ratchets: speed of the center of mass Rescaling and formal asymptotic expansion: effective diffusion Results Physical interpretation Concluding remarks

77 Introduction to ratchets models Molecular motors: how to produce motion at 1µm scale? life at low Rayleigh numbers [Purcell], modelling in biology [Vale-Milligan] Physics of brownian ratchets [Reimann]: f t = f + (f ψ(t, x) )

78 Brownian ratchet and flashing ratchet Ratchet models: ψ is t-periodic f t = f + (f ψ(t, x) ) Flashing ratchet: a model case f t = ε(t) f + (1 ε(t)) (f ψ() ) Figure: Potential and Gibbs state: e ψ when ε(t) 1/2

79 Flashing ratchet: mass transport Assume now that t ε(t) is 1-periodic ε 1 if t [0, 1/2) and ε 0 if t [1/2, 1) f t = ε(t) f + (1 ε(t)) (f ψ(x) ) Figure: The mechanism of transport [Chipot-Hastings-Kinderlehrer, Kinderlehrer-Kowlczyk, JD-Kinderlehrer-Kowlczyk]

80 Ratchets: periodic solutions 1d case, no flux boundary conditions: ε(t)f x + f ψ(t, x) x = 0 on the boundary f t = ε(t) f + (f ψ(t, x) ) The solution converges to a unique time-periodic solution Figure: The periodic solution at t = 1/2 and t = 1 Mass has been transported (to the left) of the interval

81 The doubly periodic case: entropy methods Consider a solution of g t = g + (g ψ(t, x) ) x T d, t > 0 with initial datum g 0 L 1 +(T d ), g 0 L 1 (T d ) = 1 and assume that ψ is doubly periodic: ψ(t + T, x) = ψ(t, x) Conservation of mass Contraction in relative entropy [Bartier-JD-Illner-Kowalczyk] d ϕ dt T d ( g1 g 2 ) g 2 dx = ϕ T d ( g1 g 2 ) Existence of a (unique) doubly periodic solution: g 2 = e ψ / T d e ψ dx, ϕ(s) = s log s d g 1 log dt T d ( g1 g 2 ) dx C LS g 1 log T d ( g1 g 2 ( g1 g 2 ) 2 g 2 dx ) dx + (g 2 ) t g 2 L

82 The doubly periodic case and entropy methods: results Existence of a (unique) doubly periodic solution: entropy estimate + fixed-point methods: existence of a doubly periodic solution Contraction: the doubly periodic solution attracts all other solutions If there is a logarithmic Sobolev inequality, then there is an exponential convergence in L 1 (Csiszár-Kullback inequality) [JD-Kinderlehrer-Kowlczyk] [Bartier-JD-Illner-Kowalczyk]

83 Why entropy methods? Good reasons to use entropy methods Easy estimates (compared to L or L 2 / Fourier estimates) Go well with mass conservation; gradient flow structure (Wasserstein distance [Jordan-Kinderlehrer-Otto]) Robust (1): allows for not too smooth potentials Robust (2): easy to generalize to nonlinear models Robust (3): ok even if the asymptotic state is not known Give nice results in messy problems (with various time and length scales): strong two-scale convergence But require a detailed knowledge of tricky functional inequalities

84 Stochastic Stoke s drift / traveling ratchet Stochastic Stokes drift model f t = f xx + ( ψ (x ω t)f ) x (1) ψ (x ω t) is a traveling potential moving at constant speed ω, ψ is 1 periodic: ψ(x + 1) = ψ(x) conservation of mass: R f (t, x)dx = 1 for any t 0 Position of the center of mass: x(t) := x f (t, x)dx R There exists a drift velocity or ballistic velocity c ω such that d dt x(t) c ( ω = O e t/γ ) as t A diffusive traveling front appears: effective diffusion coefficient? asymptotic profile? Ansatz: equation in self-similar variables f (t, x) = 1 R(t) u ( log R(t), x c ω t R(t) ) with R(t) := 1 + 2t

85 Travelling fronts in stochastic Stokes drifts... unlimited motion of brownian ratchets position of the center of mass? profile of the solutions for large times? rate of convergence towards the asymptotic profile? Tools: homogenized functional inequalities (logarithmic Sobolev inequalities) cell problem time-dependent asymptotic expansions (time-dependent homogenization): effective diffusion logarithmic Sobolev inequalities control the convergence Physics: efficiency

86 Homogenized functional inequalities

87 Homogenization of a Fokker-Planck equation Consider the Fokker-Planck equation with a drift corresponding to a harmonic potential modified by a periodic perturbation in the limit ε 0 + [ ut ε = uε + x u ε + 1 ( x ) ] ε φ u ε x R d t > 0 (2) ε It has a unique stationary solution with mass 1 u ε (x) := Z 1 ε e 1 2 x 2 φ(x/ε) Homogenization and turbulence, weak 2-scale convergence: [Goudon-Poupaud] How to study the convergence of u(t, ) to u? Poincaré inequality / logarithmic Sobolev inequalities / entropy methods [Bakry-Emery, Arnold-Markowich-Toscani-Unterreiter] dµ 0 := Z 1 0 e x 2 /2 dx, dµ ε := Zε 1 e φ(x/ε) dµ 0 (x) = u (x)dx ε

88 Generalized entropies, generalized Fisher information For any p (1, 2], consider for v = u/u ε E (p) ε [u] := 1 [v p 1 p (v 1)] dµ ε p 1 R d E (1) ε [u] := v log v dµ ε R d If u is a solution of (2), v solves the Ornstein-Uhlenbeck equation [ vt ε = v ε x + 1 ( x ε ε) ] φ v Generalized Fisher information: I (p) ε [u] := p v p 2 v 2 dµ ε R d d dt E(p) ε [uε (t, )] = I (p) ε [uε (t, )]

89 The strategy of entropy methods d dt E(p) ε [uε (t, )] = I (p) ε [uε (t, )] Functional inequality: for some C (p) ε > 0 decay of the entropy 4 p C(p) ε E (p) ε [u] I (p) ε [u] E (p) ε [uε (t, )] E (p) ε [uε (0, )] e 4 p C(p) ε t t 0 Generalized Csiszár-Kullback inequalities u ε (t, ) u ε L p (R d,u ε dx) C e 2 p C(p) ε t One step further: Bakry-Emery method

90 Optimal constants u p u 2, u u 2/p Logarithmic Sobolev inequality C (1) ε := inf u 0 dµ ε a.e. u H 1 (dµ ε ) R u 2 dµ d ε ( R u 2 log R u 2 d R d u 2 dµ ε ) dµ ε Generalized Poincaré (Beckner) inequalities C (p) ε := inf u 0 dµ ε a.e. u H 1 (dµ ε ) (p 1) R d u 2 dµ ε R d u 2 dµ ε ( R d u 2/p dµ ε ) p

91 A (rough) upper estimate Assume: T d e φ(y) dy = 1 and use ue(x) = x e as a test function lim ue 2 dµ ε = lim ε 0 R d ε 0 ue 2 dµ ε = 1 R d ( Rd lim ue 2/p dµ ε = 21/p 1 Γ ε 0 π ) p lim ε 0 R d ue 2 log ue 2 dµ ε = log γ where γ is Euler s constant Lemma lim ε 0 C(p) ε κ(p) := p /p π Γ ( p )

92 An extension of Holley-Stroock s perturbation lemma Convex Sobolev inequality [Arnold et al.] or ϕ-entropy [Chafaï] [ϕ(u) ϕ(ū) ϕ (ū)(u ū) ] dµ C ϕ ϕ (u) u 2 dµ ū is the average of u with respect to dµ : ū := u dµ and let d µ be a measure which is absolutely continuous with respect to dµ e b dµ d µ e a dµ µ a.e. Lemma If ϕ C 3 is a convex positive function, then [ϕ(u) ϕ(ũ) ϕ (ũ)(u ũ) ] d µ e b a C ϕ ϕ (u) u 2 d µ Consequence : C (p) ε p 2 e Osc(φ)

93 Formal asymptotic expansion in terms of ε 0 + [ ut ε = u ε + x u ε + 1 ( x ) ] ε φ u ε ε x R d t > 0 Normalization: e φ dy = 1 and ( y ) T d T d y e φ dy = 0 Assume that the solution can be written as u ε (t, x) = u (0) ( ) t, x, x x0 ε + ε u (1) ( ) t, x, x x0 ε + ε 2 u (2) ( ) t, x, x x0 ε + O(ε 3 ) where y u (i) (t, x, y) =: v (i) (t, x, y)e φ(y) is periodic... At order ε 2 : v (0) does not depend on y ( ) y u (0) + y u (0) y φ(y) = 0

94 At order ε 1 : ( ) y u (1) + y u (1) y φ(y) = x (2 y u (0) + y φ(y)u (0)) = y φ(y) x u (0) that is ( ( y e φ(y) y v (1) + x v (0))) = 0 With v (1) (t, x, y) = x v (0) (t, x) w(t, y), w(t, y) = (w j (t, y)) d j=1 solves cell equation ( ) y e φ(y) ( y w j + e j ) = 0 Thus we have obtained that [ e φ ] y w j = e T φ(y) dy 1 e j, u (1) (t, x, y) = x v (0) (x) w(t, y)e φ(y) d

95 At order ε 0 = 1 : ( u (0) t = y e φ(y) y v (2)) ( + x x v (0) + x v (0)) e φ(y) + x (2 y u (1) + y φ(y)u (1)) ( + y v (0) y e φ(y)). Solvability condition: formally integrate with respect to y T d... v (0) t = K v (0) + (x v (0)) 1 K := e T φ(y) dy e d T φ(y) dy d

96 The solution u ε (t, x) of ut ε = uε + [x u ε + 1 ε φ( x ε) ] u ε has been written as ( ( u ε (t, x) = v (0) (t, x) + ε x v (0) (t, x) w t, x ) ) + O(ε 2 ) e φ( x ε ) ε where w is a solution of the cell problem and v (0) is a solution of a Fokker-Planck equation with diffusion coefficient K. As t u(t, x, y) = v (0) (x, t)e φ(y) (1 + O(ε)) v (0) (t, x) v (0) M x (x) = 2 e 2 K (2 π K) d/2

97 A summary u ε (t, x) L 1 L 2 t uε (x) = M 1 R e 2 x 2 φ(x/ε) R d e 1 2 z 2 φ(z/ε) dz two-scale ε 0 M e x 2 (2 π) d/2 2 e φ(y) u ε (t, x) two-scale v (0) (t, x)e φ(y) L 1 L 2 ε 0 t M e x 2 (2 π K) d/2 2 K e φ(y)

98 First result: homogenized inequalities Theorem Assume that φ is a C 2 function on T d p (1, 2] lim C ε (p) = K C (p) ε Moreover, lim ε 0+ C ε (1) [k C (1) 0, K C(1) 0 ] with k = e Osc(φ) K 1 = e φ(y) e φ(y) dy T d T d C (p) 0 = p/2 ; it is an open question to determine whether lim ε 0+ C ε (1) = K C (1) 0 or not Corollary Assume that φ is a C 2 (T d ). If u is a smooth solution of (2), A[u 0 ] u ε u ε 2 L p (R d,(u ε )1 p dx) Ae 4 C(p) ε t/p t > 0 lim ε 0+ 4 C (p) ε /p = 2 K < 2 if p (1, 2], lim ε 0+ 4 C (1) ε 2 K < 2

99 Two-scale convergence Proposition ([Allaire] definition of two-scale convergence ) Let Ω be an open set in R d. If (u ε ) ε>0 is a bounded sequence in L 2 (Ω), then there exists u 0 L 2 (Ω T d ) such that, up to subsequences, ( lim u ε (x)ϕ x, x ) dx = u 0 (x, y)ϕ(x, y) dx dy (3) ε 0 Ω ε Ω T d for all smooth y periodic function ϕ. Moreover, (u ε ) ε>0 weakly converges in L 2 (Ω) to u (x) := T d u 0 (x, y) dy and lim u ε L2 (Ω) u 0 L ε 0 2 (Ω T d ) u L2 (Ω). Proposition ([Allaire]) Assume that u ε u in H 1 (Ω). Then there exist a subsequence of (u ε ) ε>0, still denoted (u ε ) ε>0, which two-scale converges to u = u (x) Moreover, there exists a function u 1 L 2 ( Ω, H 1 (T d ) ) such that u ε two-scale ε 0 x u (x) + y u 1 (x, y)

100 A compactness property Lemma The embedding H 1 (R d, dµ 0 ) L 2 (R d, dµ 0 ) is compact Proof. u n 2 H 1 (R d,dµ 0) 1, u n := v n / µ 0 u n 2 dµ 0 = v n 2 dx + 1 x 2 v n 2 dx d R d R 4 d R 2 v n 2 L 2 d u n 2 log u n 2 dµ 0 = v n 2 log v n 2 dx+log Z 0 v n 2 L 2+1 x 2 v n 2 dx R d R 2 d R d... logarithmic Sobolev inequality, with C (1) 0 = 1/2 ( u 2 dµ 0 C (1) 0 u 2 u 2 ) log dµ R d R u d R 2 0 u H 1 (R d, dµ 0 ) dµ d 0 Conclusion by Dunford-Pettis theorem Also valid for dµ ε... uniformly with respect to ε

101 Interpolation between the Poincaré and the logarithmic Sobolev inequality Lemma For any p [1, 2], C ε (p) p 2 C(2) ε Theorem ([Beckner, Arnold-Bartier-JD]) p (1, 2] C ε (2) 1 [ 1 p 1 ( 2 p p ) α ] C ε (p) with α := C(2) ε 2 C ε (1) Corollary If C ε (1) = 1 2 C(2) ε, then C (p) ε = p 2 C(2) ε for any p [1, 2]

102 The case p = 2 (1/3) There is a non-trivial minimizer u ε to C ε (2) such that R u d ε dµ ε = 0, u R d ε 2 dµ ε = 1 and ( e 1 2 x 2 φ(x/ε) u ε (x) ) = C (2) ε u ε (x)e 1 2 x 2 φ(x/ε). Let ϕ D(R d ) and ϕ 1 D(R d, C (T d )) ( x u ε [ x ϕ(x) + ε x ϕ 1 x, x ) ( + y ϕ 1 x, x )] dµ ε R ε ε d ( = C ε [ϕ(x) (2) + ε ϕ 1 x, x )] dµ ε ε R d T d R d u ε two-scale u ε x u (x) + y u 1 (x, y), K (2) ε 0 0 := lim ε 0+ C ε (2) a two-scale homogenized equation: [ ][ ] x u (x)+ y u 1 (x, y) x ϕ(x)+ y ϕ 1 (x, y) e 1 2 x 2 φ(y) dx dy = K (2) 0 R d T d u (x)ϕ(x)e 1 2 x 2 φ(y) dx dy

103 The case p = 2 (2/3) An evaluation with ϕ = 0 shows that u 1 is given as a solution of [ ] y e φ(y) ( y u 1 (x, y) + x u (x)) = 0, u 1 (x, y) = x u (x) w(y) where w = (w j ) d j=1 is the solution of the cell equation [ e φ ] y u 1 (x, y) = T e φ(y) dy 1 x u (x) d Test with ϕ = u (up to an appropriate regularization) and ϕ 1 = 0 Rd x u 2 T e φ(y) dy dµ 0 = K (2) 0 u 2 dµ 0 d R d We can also observe that R d u dµ 0 = lim ε 0 + R d u ε dµ ε = 0

104 The case p = 2 (3/3) Altogether this proves that K (2) 0 C (2) 0 e T φ(y) dy = K C(2) 0 d To prove the reverse inequality, K (2) 0 K C (2) 0, consider ( x ũ ε (x) := ue(x) + ε x ue(x)w ε) where ue(x) = x e and w is the solution to the cell problem K (2) ũ R 0 lim d ε 2 dµ ε ε 0 + ũ R d ε 2 dµ ε ( ) 2 = K C (2) 0 ũ R d ε dµ ε This completes the proof in case p = 2

105 The case p (1, 2) A result inspired by [Rothaus] Proposition Let φ be a continuous function on T d and take p (1, 2), ε > 0. Either C (p) ε p 2 C(2) ε is achieved by some non trivial function, or C (p) ε = p 2 C(2) ε is not achieved by any non trivial function No minimizer: result follows from the case p = 2 Otherwise, consider sequences of minimizers and apply the two-scale convergence approach

106 Traveling and tilted ratchets: speed of the center of mass

107 Stochastic Stoke s drift / traveling ratchet and tilted ratchet models If f is a solution of (1) f t = f xx + ( ψ (x ω t)f ) x we observe that f (t, x) = f (t, x ω t) is a solution of f t = f ( xx + (ω + ψ ) f ) a problem which is known as the tilted Smoluchowski-Feynman ratchet Tilted Brownian ratchets are actually much more general, since in the equation for f, ψ may still depend on t (flow reversals) x

108 Tilted potential Figure: x ω x + ψ(x), ψ(x) = cos x, and left: ω = 0.5, right ω = 1.25 The critical tilt: corresponds to ω = 1

109 The doubly periodic case Instead of considering (1) f t = f xx + ( ψ (x ω t)f ) x, consider g t = g xx + ( g ψ (x ω t) ) x x S 1, t > 0, g(t = 0, x) = g 0 (x) = k Z f 0 (x + k) x S 1, (4) for which, by linearity of the equations, we get g(t, x) = k Z d f (t, x + k) (t, x) R + S 1 With R f d 0 dx = 1 = R f (t, ) dx = 1 for any t 0, we can define the d position of the center of mass by x(t) := x f (t, x) dx R d

110 Speed of the center of mass d x dt = x f t dx = d ψ (x ω t)f (t, x) dx R d R d = d ψ (x ω t)f (t, x + k) dx k Z S 1 = d ψ (x ω t)g(t, x) dx S 1 t d S 1 ψ (x ω t)g (t, x) dx =: c ω The time-periodic solution g (t, x) = g ω (x ω t) solves the equation with periodic boundary conditions (g ω ) xx + ( (ω + ψ )g ω )x = 0

111 Speed of the center of mass Take a primitive: x (g ω ) x + (ω + ψ )g ω =: A(ω) is constant and ω c ω = ω 1 0 g ω dx ψ g ω dx = A(ω) Some elementary but tedious computations show that c ω < ω, lim ω 0+ c ω /ω > 0, c ω is positive for large values of ω, and lim ω c ω = 0

112 Two potentials Figure: Plots of the potential ψ(x) = sin x (left) and ψ: asymmetric smooth sawtooth potential (right)

113 Velocity of the center of mass Figure: Plots of c ω and c 0 ω (no diffusion) as functions of ω in the sinusoidal case (left) and in the case of the asymmetric smooth sawtooth potential (right, in logarithmic coordinates). In the sinusoidal case, the symmetry is reflected by the fact that c ω = c ω (values corresponding to ω < 0 are not represented). This is not true in the sawtooth case. A characteristic property of the curve ω cω 0 is the critical tilt, which is still present when diffusion is added

114 Rescaling and formal asymptotic expansion: effective diffusion

115 Heuristics If f is a solution of (1) f t = f xx + ( ψ (x ω t)f ) x and f (t, x) = 1 ( R(t) u log R(t), x c ) ω t R(t) with R(t) := 1 + 2t In the new variables: R = e t and z = R x 1 2 (R2 1)(ω c ω ), We will prove that u t = u xx + (x u) x + R ( (ψ (z) + c ω )u ) x u(t, x) = g ω (z)h 1 R g(1) ω (z)h x + O ( R 2) g ω depends only on the fast oscillating variable z, κ ω is an effective diffusion constant h t = κ ω h xx + (x h) x

116 Formal asymptotic expansion (1/2) u t = u xx + (x u) x + R [( ψ ( R x 1 2 (R2 1)A(ω) ) + c ω ) u ]x, R(t) = e t A(ω) 0... two-scale function U u(t, x) = U(t, x; z) with z := R x 1 2 (R2 1)A(ω), U t U xx (x U) x = R (U 2 zz + ( (ω + ψ (z))u ) ) + R (2 U z z + ( ) ) ψ (z) + c ω U We will formally solve the equation order by order. (i) To cancel the terms of order R 2 in the equation, U has to be proportional to g ω U(t, x; z) = g ω (z)h(t, x) + R 1 U (1) (t, x; z) + O ( R 2 ) (ii) At order R, we find that where g (1) ω (g (1) ω ) zz + U (1) (t, x; z) = g (1) ω (z)h x (t, x) is given as a solution of the equation ( (ω + ψ (z) ) ) g ω (1) z = 2 (g ω) z ( ψ (z) + c ω ) gω

117 Formal asymptotic expansion (2/2) There is a solvability condition at order R : the average on (0, 1) of the right hand side of the equation is 0. Since all functions are periodic and 1 0 g ω(z) dz = 1, we recover the condition 1 0 ψ (z)g ω (z) dz + c ω = 0 Uniqueness under proper normalization 1 0 g(1) ω (z) dz = 0 (iii) At order R 0 = 1, the solvability condition is: h t h xx (x h) x = h xx 1 0 ( ψ ) (z) + c ω g (1) ω (z) dz Hence we obtain a modified Fokker-Planck equation h t = κ ω h xx + (x h) x where the effective diffusion coefficient is given by κ ω := ψ (z)g (1) ω (z) dz

118 The effective diffusion coefficient is positive Lemma Let χ be the unique periodic solution of such that 1 0 χ dz = 0. Then κ ω ω=0 = χ (ψ + ω) χ = ψ + c ω κ ω = χ 2 g ω dz > 0 ( 1 0 eψ dz 1 0 e ψ dz) 1 = K 1 < 1 and lim ω κ ω = 1 A proof by duality [Goudon-Poupaud]