Entropy method and the large-time behavior of parabolic equations

Transcription

1 Lecture notes for XXVII Summer school in mathematical physics Ravello, Italy, September 9-1, 00 Entropy method and the large-time behavior of parabolic equations Anton Arnold Institut für Numerische und Angewandte Mathematik Universität Münster Einsteinstr. 6 D Münster, Germany. nd October arnold/ anton.arnold@tuwien.ac.at

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3 Contents 1 Large-time behavior of heat and linear Fokker-Planck equations 5 Entropy method for symmetric diffusion equations & non-symmetric extensions 9 3 Logarithmic & convex Sobolev inequalities 15 4 Pertubation results & application to semi-linear models 1 5 Entropy method for quasi-linear equations 5 6 Diffusion equations as gradient flows 9 7 Bibliography 35 3

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5 1 Large-time behavior of heat and linear Fokker-Planck equations Motivation: understand the large-time asymptotics of PDEs modeling thermodynamical systems prove convergence (with explicit rates) towards thermodynamical equilibrium study the (monotonous!) time evolution of a Lyapunov functional (physical entropy of the system) Example 1.1: kinetic (Vlasov-) Fokker-Planck equation (VFP) models time evolution of particle system (gas, plasma, electrons,...) f(x,v,t)... distribution function in the particle phase space: x Ω IR n (position),v IR n (velocity),t 0 (time);n = 1,, 3 here only Ω = IR n (= whole space problems) Figure 1.1: kinetic phase space f(x,v,t)dxdv = densitiy of particles in the volume element dxdv centered at (x,v) f(.,.,t),t 0 is a probability distribution, i.e. f 0,f(.,.,t) L 1 (IR n ), IR n f(x,v,t)dxdv = 1 (mostly assumed) 5

6 1 Large-time behavior of heat and linear Fokker-Planck equations collisions only modeled in a diffusive approximation, or modeling interaction with an environment (e.g. heat bath) [R] f t + v x f }{{} free transport x V v f }{{} influence of potential V (x,t)» v +V (x) steady state: f (x,v) = e ν σ V (x)... given confinement potential = σ v f }{{} + ν div v (vf) }{{} diffusion, σ 0 friction, ν 0 self-consistent model Vlasov-Poisson-Fokker-Planck (VPFP): (1.1) V (x,t) =V 1 (x) + Φ(x,t), e.g. V 1 (x)= x Φ =± f(x,v,t)dv = ±n(x,t) 0... spatial particle density IR n sign in Poisson equation: + for Coulomb interaction, - for gravitational case Convergence results: f(t) t f for VPF(P), no rate [BD, BCS] VFP: algebraic convergence rate [DV] exponential convergence rate for 1. V (x) = x (by Fourier techniques) [R]. space-homogeneous case, i.e. f = f(v,t) [T, AMTU] VFP: exponential convergence rate for V (x) c 0 + c 1 x (by hypoelliptic techniques) [HN] Example 1. Fokker-Planck type equations: { ρt =div(d( ρ + ρ A(x))) = div(de A ρ ), e A ρ(x,t = 0)=ρ I (x) L 1 +(IR n ) x IR n, t > 0 (1.) A(x)... given confinement potential In (1.) x may be a position vector or a velocity (e.g. in case of space-homogeneous versions of (1.1)). D = D(x)... symmetric, locally uniformly positive definite diffusion matrix. steady state: (x) = e A(x) assume: e A(x) L 1 +(IR n ); IR n e A(x) dx = 1 (w.r.o.g.) mass conservation: IR n ρ(x,t)dx = ρ I (x)dx assume (w.r.o.g.): IR n ρ I (x)dx = 1 6

7 positivity: ρ I (x) 0 ρ(x,t) > 0 x IR n,t > 0 (by a maximum principle) assume D(x), A(x) are smooth, such that a unique, global, smooth solution of (1.) exists. Questions: ρ(t) t? in which norms/topologies? convergence rate? Spectral method: symmetrization of (1.) (on L (IR n )): transformation z := ρ/ { zt =div(d z) V (x)z, x IR n,t > 0 z(t = 0)=z I := ρ I ρ, assume z I L (IR n ) with V (x) = 1 [ ( ) Tr D A 1 ] x ( A)T D A + (div D) A the Hamiltonian Hz = div(d(x) z) + V (x)z satisfies [ < Hz,z >= T z D(x) z + V (x)z ] dx = IR n σ(h) [0, ) IR n T < Hz,z >= 0 z = C, C IR ground state z = e A of H is non-degenerate is unique normalized steady state of (1.) Spectral gap: ( z ρ )D(x) sufficient condition for the case D(x) I (identity matrix): V L 1 loc,v (x) bounded below, V (x) for x [RS]; (example: A(x) = c x α,α > 1) positive spectral gap λ 0 > 0 ρ(t) L (IR n,ρ 1 ) z(t) L (IR n ) z I L (IR n ) e tλ 0 example: A(x) = c x λ 0 = c L 1 -convergence: ρ(t) dx = IR n IR n ( z ρ ) (x)dx 0 ρ ρ(t) ρ dx = O ( e tλ 0 ) (1.3) 7

8 1 Large-time behavior of heat and linear Fokker-Planck equations Drawbacks of spectral method ([AMTU, MV]): unnatural constraint ρ I ρ L (IR n ) explicit values / estimates for λ 0 hard to get method hard to generalize to non-linear problems Example 1.3: heat equation { uτ =D ξ u, ξ IR n, τ > 0; D > 0 u(ξ,τ = 0)=u I (ξ) 0; u IR n I (ξ)dξ = 1 has only trivial steady state, but τ-dependent asymptotic state for τ time-dependent rescaling transformation to Fokker-Planck equation ([T1, CT, AMT]): ( ) ξ u(ξ,τ) = R(τ) n v, ln R(τ) (1.4) R(τ) with R(τ) = Dτ + 1, v = v(x,t); t = lnr(τ) satisfies: { vt =div x ( x v + xv), x IR n,t > 0 v(x,t = 0)=v I (x) = u I (x) steady state of v: v (x) = (π) n e x (1.5) asymptotic state of( u (from ) 1.4): u as (ξ,τ) = R(τ) n ξ v... spreading Gaussian; fundamental solution of heat R(τ) equation convergence (from (1.3) with A(x) = x, λ 0 = 1): Exercise: v(t) v L 1 (IR n x ) Ce t u(τ) u as (τ) L 1 (IR n ξ ) C(Dτ + 1) 1 1. Find a rescaling of form (1.4) to transform u τ = (u m ), ξ IR n,τ > 0 (fast diffusion equation if 0 < m < 1, porous medium equation if m > 1) into v t = (v m ) + div(xv), x IR n,t > 0 with u I = v I.. Calculate the spectral gap λ 0 corresponding to A(x) = x (in 1D). Hint: The eigenfunctions of the harmonic oscillator Hamiltonian +x are H n (x)e x /, with the Hermite polynomials H 0 = 1, H 1 = x, H n+1 = xh n nh n 1. 8

9 Entropy method for symmetric diffusion equations & non-symmetric extensions Example.1: space-homogeneous BGK relaxation model [BGK] for f(v,t), v IR n,t > 0: { ft =λ(m[f] f), t > 0, λ 1... f(v,t = 0)=f I (v) relaxation time { } M[f](v) = m(πθ) n v u exp... Θ... Gaussian with mass m, mean velocity u, temperature Θ IR n (.1) m[f], u[f], Θ[f] are chosen such, that 1 1 v fdv = v M[f]dv (.) v v IR n from simple calculation: m,u, Θ are const. in t (.1) is a linear inhomogeneous ODE steady state: f = M[f I ] from explicit solution f(v, t) 0: f(t) M[f I ] L 1 (IR n ) = ce λt (.3) Entropy approach (first idea): alternative method for convergence f(t) t f = M[f I ] relative entropy of state f(t) w.r.t. f : e(f(t) f ) := f(t) ln f(t) IR n f dv = ( ψ f(t) IR n f )f dv 0 with ψ(σ) := σ ln σ σ + 1 0, σ 0 (note: f(v,t)dv = f I (v)dv = f (v)dv; from (.1), (.)) 9

10 Entropy method for symmetric diffusion equations & non-symmetric extensions entropy dissipation: I(t) := dt d e(f(t) f ) = λ ( ) ( ) f f IR n f 1 ln f f dv λe(f(t) f ) 0 (.4) (use (.1) and (σ 1) ln σ σ ln σ σ + 1) e(f(t) f ) e λt e(f I f ), t 0 use Csiszár-Kullback inequality [Cs, K, UAMT]: f 1 f L 1 (IR n ) e(f 1 f ) (.5) f 1,f L 1 +(IR n ) with f 1 dv = f dv = 1. f(t) f L 1 (IR n ) e(f I f ) e λ t suboptimal rate (cp. (.3)) we shall later use a differential inequality between I (t) and I(t) (instead of e (t) and e(t) in (.4)) Admissible relative entropies: Definition.: let J = IR + or IR, let ψ C( J) C 4 (J) with ψ 0, ψ(1) = 0, ψ > 0, (ψ ) 1 ψ ψ IV (.6) ρ 1 L 1 (IR n ), ρ L 1 +(IR n ) with ρ 1 dx = ρ dx = 1 e ψ (ρ 1 ρ ) := ( ρ1 ψ ρ IR n ) ρ dx 0 is an admissible relative entropy (of ρ 1 w.r.t. ρ ) with generating function ψ. Examples: 1. ψ 1 (σ) = σ ln σ σ + 1 on J = IR + (physical entropy). ψ p (σ) = σ p 1 p(σ 1), 1 < p on J = IR + (or J = IR for p = ), e p (ρ 1 ρ ) := e ψp (ρ 1 ρ ). note: e (ρ 1 ρ ) = ρ 1 ρ L (IR n ;ρ 1 ) e ψ is a convex functional of ρ 1 10

11 Entropy dissipation for Fokker-Planck type equations: rewrite (1.) as: ( ) ρ ρ t = div( Du), u :=, = e A(x) (.7) entropy dissipation: I ψ (ρ(t) ) := dt d e(ρ(t) ) = ( ψ IR n = IR n ψ ( (remember: ψ > 0, D > 0) ρ ρ ) ρ t dx ) u T Du dx 0 entropy dissipation rate: here for special case D(x) I (identity matrix); extensions: D(x) = D(x)I (scalar diffusion) [AMTU]; D(x) = matrix [BE1, BE]. d dt I ψ(ρ(t) ) = ( ) ρ ψ u T A x udx + Tr(XY ) dx } {{ } 0 (.8) Since ( ψ ( ρ ρ X = ) ψ ( ρ ψ ( ρ 1 ) ψiv ( ρ ) ) ) 0 by (.6) Y = ( ij ( u i x j ) u T u u x u T u x u u 4 ) 0 Remarks: the Jacobian u is symmetric x for ψ = ψ p (with 1 < p < ) X > 0 holds analysis can be refined [AD] Bakry-Emery-condition: [BE1] assume for A(x),D(x): There exists a λ 1 > 0 such that either: 11

12 Entropy method for symmetric diffusion equations & non-symmetric extensions D I, A(x) x λ 1 I x IR n (uniform strict convexity) (.9) or or D(x) = D(x)I (scalar diffusion), ( 1 n ) 1 4 D D D + 1 ( D D A)I +D A A D + D A + D x x λ 1I (.10) D(x) D (const. matrix), A(x) x λ 1 D 1 (cf. [AMTU]) (.11) Exponential decay for entropy dissipation: let D I, A(x) x λ 1 I (.8) I ψ λ 1 I ψ (.1) I ψ (t) e λ 1t I ψ (t = 0), t 0 (.13) Exponential decay of entropy: integrate (.1) from t to e ψ(t) = I ψ (t) λ 1 e(t) (.14) Theorem.: Let D(x),A(x) satisfy (.9), (.10), or (.11). Let e ψ (ρ(t) ) < e ψ (ρ 0 ) e ψ (ρ I )e λ 1t, t > 0 Remark for proof: above calculation requires I ψ (ρ I ) <, can be eliminated by a density argument [AMTU] Remark: a generalized Csiszár-Kullback inequality (cf. (.5); [AMTU, UAMT]) yields L 1 -decay: ρ(t) L 1 (IR n ) e λ 1t ψ (1) e ψ(ρ I, ), t > 0. Non-symmetric Fokker-Planck equations: { ρt =div(d( ρ + ρ( A + F))), x IR n, t > 0 ρ(t = 1)=ρ I Assume again ρ I L 1 +(IR n ), ρ I dx = 1; D(x),A(x) satisfy (.9), (.10), or (.11). Let n, otherwise nothing new. (.15) 1

13 let the pertubation F = F(x,t) satisfy div(d F ) = 0 on IR n (0, ) (.16) = e A(x) is still a steady state; the operator div(dρ F) is skew-symmetric in L (IR n,ρ 1 ) relative entropy: e ψ (ρ(t) ) := ψ entropy dissipation: d dt e ψ(ρ(t) ) = ( ρ(t) ) dx ( ) ρ(t) ψ u T Du dx ρ } {{} =I ψ (ρ(t) ) ( ) ρ(t) ψ div(dfρ ρ )dx (.17) } {{} =0 by (.16) e ψ and e ψ are independent of F! use inequality (.14) for (.17): d dt e ψ(ρ(t) ) λ 1 e ψ (ρ(t) ) e ψ (ρ(t) ) e λ 1t e ψ (ρ I ), t 0 i.e. same decay estimate as for symmetric equation (1.) [AMTU, AC] Example.3: quantum Fokker-Planck equation for quadratic potential V = V (y): f t + v y f y v f = Lf; y,v IR d, t > 0 (.18) Lf = D pp v f + γ div v (vf) + D qq y f + D pq div y ( v f) and the constants satisfy γ > 0, D pp D qq D pq γ 4, D pp > 0,D qq > 0 unique normalized steady state: Exercise: γ f (y,v) = (π) d exp( Q(y,v)), σ Q(y,v) := γ [(D σ pp + D qq ) y + 4γD qq (y v) + (D qq + D pp + 4γ(D pq + γd qq )) v ], σ := Dqq + Dpp + D qq D pp (1 + γ ) + 4γD pq (D qq + D pp ) 1. Rewrite (.18) in form (.15), (.16) with x = (y,v) IR d. Let f I (y,v) L 1 +(IR d ) with f I dydv = 1. Then prove that e ψ (f(t) f ) decays exponentially.. Derive the condition (.11) from (.9): use the coordinate transformation y(x) = D 1 x to transform (1.) (with constant matrix D) into a FP-type equation with the identity as diffusion matrix. 13

14 Entropy method for symmetric diffusion equations & non-symmetric extensions 3. Calculate the entropy decay rate for the classical FP-equation ρ t = div( ρ + xρ), x IR n. (.19) Show ρ(t) in L 1 (IR n ) (with exponential rate) also for ρ I with ρ I dx = 1 but not necessarily ρ I (x) 0. 14

15 3 Logarithmic & convex Sobolev inequalities Standard Sobolev inequality: f L p (IR n ) C n f L (IR n ) f H 1 (IR n ); n 3 (3.1) < p = n n (no additional information on local singularities / integrability n survives as n ) Convex Sobolev inequalities (CSI): Consider (.14) for D I (under assumption (.9)): e ψ (ρ ) 1 λ 1 I ψ (ρ ) i.e. CSI in entropy version: IR n ψ ( ρ ) dx 1 λ 1 IR n ψ ( ρ ) ρ dx (3.) ρ, L 1 +(IR n ) with ρdx = dx Example 3.1: logarithmic Sobolev inequality (LSI): let ψ = ψ 1 (σ) = σ ln σ σ + 1 ρ ln ρ dx 1 λ 1 ρ ln ρ dx use transformation f = ρ : IR n f ln f dx 1 λ 1 LSI in steady state measure version f dx f L +(IR n ; ), f dx = dx (3.3) 15

16 3 Logarithmic & convex Sobolev inequalities particular case for A(x) = 1 a x, a > 0 λ 1 = 1, a (x) = (πa) n exp( x ) =: M a a(x)... Gaussian f ln fm a dx a f M a dx f M a dx = 1 (3.4) IR n IR n LSI of Gross [G1] comparison of (3.1), (3.4): l.h.s. of (3.4) may be negative, but it is bounded below (f ln f C,M a (x)dx is a bounded measure) f ln f provides less information on local singularities than f p ;p >, but it survives as n the constant a in (3.4) is independent of n; C n is not (3.4) is a non-linear inequality makes the normalization f M a dx = 1 necessary (otherwise additional terms appear in (3.4), see Exercise 1) Example 3.: CSI for power law entropies: let ψ = ψ p (σ) = σ p p(σ 1) 1, 1 < p use transformation ρ = f p R f p dx in (3.); assume dx = 1 p [ p 1 ( f dx ) p ] f p ρ dx λ1 f dx f L p (IR n ; ) (3.5) Beckner inequalities [B, AMTU] special case p = (Poincaré / spectral gap inequality (PI)): ( ) f dx f dx f dx f L 1 (IR n ; ) (3.6) λ 1 Remarks: The p 1 limit of the inequalities (3.5) yields the LSI (3.3) (without the normalisation constraint) among the CSI s (3.3), (3.5) the LSI (3.3) is the strongest and the PI (3.6) weakest, in the following sense: (3.3) (3.6) f L (IR n ; ) L 1 (IR n ; ) The PI (3.6) implies that the Hamiltonian H = + V (x) of 1 (on L (IR n )) has a spectral gap λ 1 [G],[AMTU]. Hence, λ 1 λ 0, where λ 1 is the best constant in (3.3) (logarithmic Sobolev constant) and λ 0 the spectral gap. 16

17 sharpness of CSI s: question: For which ρ does the CSI (3.) become an equality? [Ca, T3, AMTU] recall derivation of CSI for D I (in (.8)-(.1)): d dt I ψ(ρ(t) ) = λ 1 I ψ (ρ(t) ) + r ψ (ρ(t)), with remainder r ψ (ρ(t)) 0 integrating w.r.t. t: e ψ (ρ I ) = 1 λ 1 I ψ (ρ I ) 1 λ 1 0 r ψ (ρ(s))ds The CSI e 1 λ 1 I is an equality iff r ψ (ρ(t)) = 0 for a.e. t (0, ), i.e. IR n ψ ( ρ ) ( ) u T A x λ 1I uρ dx + for trajectory ρ = ρ(t) with u = ρ. conditions for (3.7): IR n Tr(XY ) dx = 0 (3.7) detx = 1 ( ) ( ) ( ) ρ ρ ρ ψ ψ IV ψ = 0 (3.8) dety = u ( ) 4 ui (u T u x i,j j x u) = 0 (3.9) ( ) ρ ψ u T u ( ) ρ x u + ψ u 4 = 0 (3.10) ( ) u T A x λ 1I u = 0 (3.11) saturation only for logarithmic or quadratic entropies (check (3.8) for ψ p, 1 p ) Theorem 3.3: For D I and ψ(σ) = σ ln σ σ + 1, the LSI (3.3) is an equality iff: 1. A(x(y)) = λ 1 y 1 + βy 1 + B(y,...,y n ) (3.1) for some Cartesian coordinates y(x) = (y 1,...,y n ) and some β IR; 17

18 3 Logarithmic & convex Sobolev inequalities. ρ = exp [ A(x(y)) + ξy 1 ξ + βξ ] λ 1 λ 1 for some ξ IR. (3.13) Proof: Using ψ 1 and z := ln( ρ ) in (3.7) gives r ψ1 (ρ) = ( ) ρz T A x λ 1I zdx + ρ i,j ( ) zi dx. x j Since z 0 z 0 z(x,t) C(t) IRn x ρ(x,t) = C 1 (t) exp[c(t) x A(x)]; C 1 (t) > 0 FP equation (.7) [ C 1 + C 1 x C = C 1 C ] C 1 C A(x) ρ(x,t) = exp (C 0 xe λ1t + α λ 1 )e λ1t A(x) with (A(x) λ 1 x ) C 0 = α qed. Theorem 3.4: For D I and ψ(σ) = (σ 1), the CSI (3.) is an equality iff: 1. (3.1) holds;. ρ = (1 + ξy 1 ) exp[ A(x(y))] (3.14) for some ξ IR. Note: For p =, ρ < 0 is allowed. (3.13), (3.14) are extremal functions 18

19 Example 3.5 (exp. decay of extremal functions): A(x) = x, n = 1 1. extremal function for ψ 1 (σ) = σ ln σ σ + 1: ρ 1 I (x) = + x] > 0 exp[ x 10 time evolution of logarithmic ( ) and quadratic ( ) relative entropy e(ρ n ) n... time steps Figure 3.1: decay of relativ entropies. extremal function for ψ (σ) = (σ 1) : ρ I (x) = (1 + x) ] IR exp[ x extremal function as ρ I slowest decay: e t for large times numerical effects are visible Exercise: 1. Derive the LSI (3.3) (without the f dx = dx constraint) in the p 1 limit of (3.5).. Prove: LSI PI. Hint: Consider the ǫ 0 limit of LSI for f (x) = 1 + ǫg(x); g L (IR n ; ). 3. Prove Theorem 3.4 (1D-proof is easier). 19

20 3 Logarithmic & convex Sobolev inequalities 10 0 time evolution of logarithmic ( ) and quadratic ( ) relative entropy e(ρ n ) n... time steps Figure 3.: decay of relativ entropies 0

21 4 Pertubation results & application to semi-linear models Example 4.1: A(x) = x 4, x IR; D I violates the BE-condition A (x) λ 1 > 0 at x = 0. But its FP-operator has a positive spectral gap; also: exponential decay of relative entropy 1 quartic potential 0.5 A(x), A ~ (x) x Figure 4.1: Ã(x) = x 4 ( ) is a bounded perturbation of the uniformly convex function A(x) (- - -) Holly-Stroock-pertubation [HS, AMTU]: Theorem 4.: Let (x) = e A(x), D(x) satisfy the CSI (3.) (use ρ = f f L (ρ )): ψ ( f f L () ) dx λ1 ( ) f ψ f T f D fρ f 4 L () f dx f L ( ), L () (4.1) for some ψ = ψ p (σ) 0. Let (x) = e Ã(x) with dx = dx = 1, Ã(x) = A(x) + v(x), 0 < a e v(x) b <, x IR n, (4.) 1

22 4 Pertubation results & application to semi-linear models i.e. Ã(x) is an L (IR n )-perturbation of A(x). Then also satisfies a CSI of form (4.1), where replaces and the constant is λ 1 := a b λ 1 < λ 1. A drift-diffusion-poisson model: ( ( ) ρ t =div ρ + x + V (x,t) ρ(x,t = 0)=ρ I (x) 0; ρi dx = 1 V (x,t) = 1 4π ) ρ, x IR 3, t > 0 ρ(y,t) x y dy... solves V = ρ (4.3) model for electron transport in plasma, semiconductor (external) confinement potential V ext = x can be removed by time-dependent rescaling (cf. Example 1.3; [AMT]) unique normalized steady state satisfies the mean-field equation, a semilinear elliptic equation for V : (x) := V (x) = 1 4π» exp x V (x) h y R IR 3 exp V (y) i IR 3 (y) x y dy dy = V relative entropy-type functional: ( ) ρ e(ρ) := ψ 1 dx + 1 (V [ρ] V ) dx (4.4) IR 3 IR 3 Theorem 4.3 [AMTU]: Let ρ I L 1 +(IR 3 ) L 3 +ǫ (IR 3 ) λ 1 > 0 such that Proof: e(ρ(t)) e λ 1t e(ρ I ), t 0; V (t) V L (IR 3 ) = O d dt e(t) = ρ(t) ρ(t) ln N(t) IR 3 ( ) e λ 1 t. dx, (4.5) with t-local equilibrium state: ] ] N(t) := exp [ x V (x,t) / exp [ y V (y,t) IR 3 dy

23 since V (t) L (IR 3 ) K, t > 0 (cf. [AMTU]): V (x,t) is a bounded perturbation (uniformly in t 0) of the uniformly convex potential x a LSI holds for the potential x +V (x,t) by Theorem 4.: λ 1 > 0 (independent of t) with: ρ ln ρ N dx 1 λ 1 ρ ln ρ dx, N ρdx = 1 d e(t) dt λ 1 ρ ln ρ N dx = λ 1 ρ ln ρ dx λ 1 (V (t) V )ρ dx λ 1 ln ( e V V (t) dx ) λ 1 ρ ln ρ dx λ 1 (V (t) V ) (ρ(t) ) dx (use Jensen s inequality since ln σ is convex) use Poisson equation: ρ(t) = ((V (t) V ) d dt e(t) λ 1 e(t) qed. Philosophy of proof: use LSI for e differential inequality for e(t) Coupled oscillator model [AMTU, ABM]: ρ t = div ( ρ + [ A(x) + Θ z ρ(t) x ] ρ ),x IR n,t > 0 z ρ (t) = xρ(x,t)dx IR n ρ(x, 0)=ρ I (x) L 1 +(IR n ), ρ I dx = 1 (4.6) collective dynamics of spin systems, muscle fibers,... t-local equilibrium state: ρ 0 (x;ρ(t)) := exp [ A(x) Θ z ρ(t) x ]... not normalized. relative entropy -type funcitional: e(ρ ρ 0 ) := IR n ρ ln ρ ρ 0 dx... bounded below, but not 0. entropy dissipation: I(ρ ρ 0 ) = IR n ρ ln ρ ρ 0 dx 0 entropy dissipation rate: d I(ρ ρ dt 0) = λi(ρ ρ 0 ) + r(t) A λi > 0 r(t) 0 x exponential decay: I(ρ ρ 0 )(t) e λt I (ρ I ρ 0 (t = 0)) 3

24 4 Pertubation results & application to semi-linear models exponential convergence of first moment: ż ρ (t) = ( )1 ρ ln ρ ρ 0 dx ρ ln ρ ρ 0 dx = I (ρ ρ 0 ) 1 Hölder z ρ (t) z ρ ( ) = O(e λt ); z ρ ( )...unknown (4.7) steady states (possibly not unique): (x;ρ(t)) :=... normalized ρ 0(x;ρ(t)) R ρ0 (y;ρ(t))dy (x;ρ(t)) t (x) exponential entropy decay e (ρ(t) (t)) ( ) 1 (λ+θ) I (ρ(t) (t)) = 1 (λ+θ) I (ρ(t) ρ 0(t)) = O ( e λt) where ( ) is the LSI for W(x) := A(x) + θ z ρ(t) x ; W x (λ + θ)i Philosophy of proof: (i) entropy dissipation method for I-decay (ii) use LSI for e e-decay Exercise: 1. Prove Theorem 4.. Hint: estimate ψ p ( f f L ( ) ) dx by using (in this order): the homogeinity of ψ p (σ), b, the CSI (4.1), 1 ρ a, and the homogeinity of ψ p(σ).. Verify that e = F(ρ) F( ) holds for (4.4), with the free energy: F(ρ) := (ρ ln ρ + x ρ + V (x) ) ρ dx (4.8) IR 3 (sum of entropy and potential energy). Remark: In (4.8) the potential energy density of the external potential V ext = x / is V ext ρ, but for the self-consistent Coulomb potential it is 1 V ρ. 3. Verify (4.5). 4

25 5 Entropy method for quasi-linear equations Example 5.1: porous medium equation [CT, MV, DdP, CMU, CJMTU]: { uτ = ξ (u m ) = m div ξ (u m 1 ξ u), ξ IR n,τ > 0,m > 1 u(ξ,τ = 0)=u I (ξ) 0, u IR n I dξ =: M > 0 models slow diffusion of gas / fluid in a porous medium (5.1) Cauchy problem for (5.1) is well-posed for any u I L 1 +(IR n ):! mass preserving solution u C([0, ),L 1 +(IR n )); the (generalized) solution u(.,t) is at least Hölder continuous (5.1) is a degenerate diffusion equation (at u = 0) it may show finite propagation speed: if u I is compactly supported, so is u(.,t) for any t > 0 propagating interfaces (free boundary) self-similar Barenblatt-Prattle solution: ( U(ξ,τ) = τ kn C 1 (m 1)k m ξ τ k ) 1 m 1 with k = (n(m 1) + ) 1, C 1 such that U(ξ,τ)dξ = M IR n U(.,τ) is compactly supported; U(ξ,τ) τ 0 M δ 0 (ξ) +, (5.) a time-dependent rescaling of the form ( ) ξ u(ξ,τ) = R(τ) n ρ, ln R(τ) R(τ) (cf. Exercise 1.1) transforms (5.1) to { ρt =div x ( x ρ m + xρ), x IR n,t > 0 ρ(x,t = 0)=ρ I (x) = u I (x) 0 (5.3) (5.4) steady state: (x) = ( C m 1 ) 1 m 1 m x, (5.5) + with C such that dx = M. 5

26 5 Entropy method for quasi-linear equations Barenblatt Prattle solution at τ 1 = 1, τ = 3, τ 3 = U(ξ,τ) ξ Figure 5.1: Barenblatt-Prattle solution for m =, M = 1 (generalized) free energy functional: E(ρ) := IR n ( x ρ + 1 ) m 1 ρm Remark: E(ρ) 1 m 1 m 1 ρdx e 1 (ρ e x / ) dx 0 (strictly convex in ρ) relative free energy: H(ρ ) := E(ρ) E( ) Lemma 5.1: ρ = is the unique minimizer of H(ρ ) under the constraints ρdx = ρ dx; ρ, 0. dissipation of free energy: d dt E(ρ) = d dt H(ρ ) =: I(ρ ) = IR n ρ x + m m 1 ρm 1 dx 0 (5.6) dissipation rate of free energy: d dt I (ρ(t) ) = I (ρ(t) ) + R (ρ(t)), (5.7) 6

27 R (ρ(t)) = (m 1) IR n ρ m [div y] dx + IR n ρ m [ n i,j=1 y = x + m m 1 ρm 1 note: x = I λ x 1 = 1 I I I(t) e t I(t = 0) exponential decay of free energy: integrate (5.8) from t to : ( y i x j ) ]dx 0, (5.8) H (ρ(t) ) = I (ρ(t) ) H (ρ(t) ) (5.9) Theorem 5. [CT, CMU]: Let ρ I L 1 + L (IR n ), x +ǫ ρ I (x) L 1 (IR n ), E(ρ I ) < Remarks: H (ρ(t) ) e t H (ρ I ), t 0 (5.10) technical difficulty in proof: above derivation requires lots of regularity of ρ(x,t) - usually does not hold! approximate ρ I (x) by smooth ρ N I (x) > 0 + density argument L 1 (IR n ) - convergence of ρ(t) to (with exponential rate) follows from (5.10) and a generalized Csiszár-Kullback inequality [CT, DdP] while exponential decay is natural for linear equations, exponential decay of (5.4) (without resorting to linearization) is a remarkable property L 1 -convergence of porous medium solution u(ξ,τ) to U(ξ,τ) follows from rescaling (5.3) (5.9) is a generalized CSI: ( n + 1 ) ρ m dx m 1 IR n Exercise: use I = x ρdx n ρ m dx + 1 ( ) 1 m m 1 IR n ( m m 1 ρ m 1 ) ρ m 1 dx 1. Calculate by setting the flux ρ m + xρ of (5.4) equal to zero. Relate (via (5.3)) (x) to U(ξ,τ) in (5.).. Verify the formula for I(ρ ) in (5.6). dx + E( ) (5.11) 3. Prove Lemma 5.1 by a variational argument with Lagrange multiplyers: For ρ 0, consider the first variation of E(ρ) + λ(m ρdx). 4. Derive the Gross-LSI from (5.11). 1 Hint: subtract m 1 ρdx from both sides, and let m 1. 7

28 5 Entropy method for quasi-linear equations 8

29 6 Diffusion equations as gradient flows consider solution of a diffusion equation (e.g. Fokker-Planck equation): ρ(t) with convex Lyapunov functional E (ρ(t)) ց Figure 6.1: The functional E(ρ) decays monotonously along the trajectories ρ(t). follows the steepest decent from ρ I to. question: find a metric on space of ρ such that the trajectory ρ(t) follows steepest decend of E(ρ): dρ dt = E(ρ) in a sense to be made precise Example 6.1: heat equation { ρt = ρ, x IR n ρi dx = 1, ρ I L 1 +(IR n ) ρ(t) P(IR n ) := {ρ L 1 +(IR n ) ρdx = 1}... space of probability densities convex functional: E(ρ) = 1 ρ dx : P(IR n ) IR + IR n 0 Frï 1het derivative: DE(ρ) = ( ρ, ) L linear functional; can be identified on L (IR n ) with ρ ρ t = E(ρ) = ρ (in sloppy notation) 9

30 6 Diffusion equations as gradient flows Definition of gradient flow [O], 8,9 of [V]: trajectory ρ(t) M = P(IR n )... manifold (due to the constraints) dρ dt T ρm = { functions s on IR n IR n sdx = 0}... tangent space 3 ingredients of a gradient flow: (i) a differentiable manifold M Figure 6.: manifold M and tangent space T ρ M (ii) a metric tensor g on M (= inner product on each T ρ M, ρ M); (M,g) is a Riemannian manifold (iii) a functional E : M IR the autonomous differential equation dρ dt g ρ : T ρ M T ρ M IR = E(ρ) (6.1) is then called a gradient flow purpose of (ii): E(ρ)... direction of strongest growth of E, when varying ρ along T ρ M - directions of same (infinitesimal) length need a notion of length within T ρ M! Like in Example 6.1, the inner product g ρ allows to identify the linear functional DE(ρ) (acting on T ρ M) with E(ρ) T ρ M: g ρ ( E(ρ),s) = DE(ρ) }{{} (s) s T ρm (6.) տ ր functional T ρm on T ρm E decays along ρ(t) ( :) d dρ E(ρ(t)) = DE(ρ) dt dt }{{} T ρm (6.) = g ρ ( E(ρ), dρ dt ) (6.1) = g ρ ( dρ dt, dρ dt) 0 30

31 Example 6.: porous medium equation as gradient flow: (i) M = P(IR n ), T ρ M = { sdx = 0} (ii) g ρ (s 1,s ) := ρ p IR n 1 p dx = s 1 p dx s 1, T ρ M with the auxiliary functions p 1, satisfying div(ρ p 1, ) = s 1, { 1 (iii) E(ρ) := m 1 (ρ m ρ)dx, m > 1 ρ ln ρdx, m = 1 { 1 DE(ρ)(s) = m 1 (mρ m 1 1) sdx, m > 1 (lnρ + 1)sdx, m = 1 (6.1) + (6.): We have s T ρ M: 0 = g ( { ρ dρ,s) + DE(ρ)(s) = p + 1 t m 1 (mρ m 1 1)sdx, m > 1 dt ρ p + (lnρ + 1)sdx, m = 1 { t ρ = p } 1 t m 1 (mρ m 1 1) div(ρ p)dx, m > 1 ρ p (lnρ + 1)div (ρ p)dx, m = 1 t = ( ρ ρm) pdx = 0, t where div(ρ p) = s is used. } Motivation for framework in Example 6.: in contrast to Example 6.1 (for m = 1) E(ρ) = ρ ln ρ has a clear thermodynamical interpretation steepest decent methods (t-semidiscretization) in variational form [JKO], e.g. d(ρ n,ρ n+1 ) min ρ n M + E(ρ n+1 ) E(ρ n ) t for d(.,.) = L ρn+1 ρ n = E(ρ n+1 ) t unified framework for ρ = div t x [ρ x U (ρ) + ρ x V + ρ(ρ x W)], x IR n,t > 0 with functional E = U + V + W, internal energy U(ρ) = U (ρ(x))dx potential energy V(ρ) = ρ(x)v (x)dx interaction energy W(ρ) = 1 W(x y)ρ(x)ρ(y)dxdy Examples: 1. U(s) = sm s m 1 x,v (x) =,W = 0,m 1... confined porous medium/fp equation. U(s) = σs log s,v = 0,W(z) = z 3 ;n = 1... granular flow in heat bath }{{} 3 σ ρ 31

32 6 Diffusion equations as gradient flows Wasserstein distance: a metric g on T ρ M induces a distance on M (geodesic length between ρ 0 and ρ 1 ): W(ρ 0,ρ 1 ) 1 = inf all paths { dρ ρ(t) dt ρ(0) = ρ 0,ρ(1) = ρ 1 } 0 }{{} dt T ρ(t) M This distance is realized along the geodesic ρ(t) conecting ρ 0 and ρ 1. Displacement convexity [MV]: Definition: The functional E : M IR is (a) displacement convex, if ρ 0,ρ 1 M : E ( ρ(t)) : [0, 1] t IR is convex (b) uniformly displacement convex, if ρ 0,ρ 1 M : d dt E ( ρ(t)) λw(ρ 0,ρ 1 ), 0 < t < 1 Examples: V(ρ), W(ρ), U(ρ), if V (x) is (uniformly) convex if W(z) is convex if U(0) = 0 and (U (ρ) ρu(ρ))ρ 1 n is non-decreasing Remark: (uniform) displacement convexity can also be defined in terms of the Hessian of E (w.r.t. (M,g)) Steady state convergence: assume that E(ρ) has a unique minimizer (in M) H(ρ ) := E(ρ) E( ) Theorem 6.3: let E be uniformly displacement convex with λ > 0 H (ρ(t) ) H(ρ I )e λt Remarks: Theorem 6.3 is purely f ormal, due to existence, uniqueness and regularity questions of ρ(x,t) but a good guideline for the rigorous statement and proofs 3

33 Exercise: 1. Verify that the metric g(s 1,s ) := p 1 p dx; p j = s j ; j = 1, IR n and the functional E(ρ) = 1 m+1 ρ m+1 dx, m > 1 constitute a gradient flow representation of the porous medium equation. Remark: The metric g is the homogeneous part of the H 1 inner product.. Verify that the metric from Example 6. and the functional E(ρ) = [ρ ln ρ + ρv (x)] dx constitute a gradient flow representation of the linear Fokker-Planck equation. 3. Verify that the metric from Example 6. and the functional E(ρ) = 1 ρ(x)ρ(y)w(x y)dxdy constitute a gradient flow representation of ρ t = div(ρ (ρ W)). 33

34 6 Diffusion equations as gradient flows 34

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37 7 Bibliography [HN] F. Hérau, F. Nier, Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential, Preprint, 00. [HS] R. Holley and D. Stroock, Logarithmic Sobolev Inequalities and Stochastic Ising Models, J. Stat. Phys. 46, No. 5-6 (1987) [JKO] R. Jordan, D. Kinderlehrer, F. Otto, The variational formulation of the Fokker- Planck equation, SIAM J. Math. Anal. 9, No. 1 (1998) [K] S. Kullback, Information Theory and Statistics, John Wiley, [MV] P.A. Markowich, C. Villani, On the trend to equilibrium for the Fokker-Planck equation: an interplay between physics and functional analysis, Matematica Contemporanea (SBM) 19 (000) [O] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. PDE 6, No. 1- (001) [R] H. Risken, The Fokker-Planck Equation, Springer Verlag, 1989 [RS] M. Reed, B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, [SCDM] C. Sparber, J.A. Carillo, J. Dolbeault. P.A. Markowich, On the Long Time Behavior of the Quantum Fokker-Planck equation, submitted, 00. [T1] G. Toscani, Kinetic approach to the asymptotic behaviour of the solution to diffusion equations, Rend. di Matematica 16 (1996) [T] G. Toscani, Entropy dissipation and the rate of convergence to equilibrium for the Fokker-Planck equation, Quart. Appl. Math. 57 (1999) [T3] G. Toscani, Sur l inégalité logarithmique de Sobolev, C.R. Acad. Sc. Paris 34 (1997) [UAMT] A. Unterreiter, A. Arnold, P. Markowich, G. Toscani, On generalized Csiszár- Kullback inequalities, Monatshefte fr Math. 131 (000) [V1] C. Villani, A review of mathematical topics in collisional kinetic theory, to appear in Handbook of Fluid Mechanics, S. Friedlander, D. Serre (Eds.), 00. [V] C. Villani, Topics in mass transportation, Preprint,