1 Entropy-dissipation methods I: Fokker-Planck equations Ansgar Jüngel Vienna University of Technology, Austria www.jungel.at.vu Introduction Boltzmann equation Fokker-Planck equations Degenerate parabolic systems
Introduction 2 Entropy = Measure of molecular disorder or energy dispersal Entropy in physics: Introduced by Clausius (1865) in thermodynamics Boltzmann, Gibbs, Maxwell: statistical interpretation Shannon (1948): concept of information entropy Entropy in mathematics: Lyapunov functional (nonincreasing along trajectories) Lax (1973): entropy in hyperbolic conservation laws DiPerna Lions (1989): entropy in kinetic theory Connections to functional inequalities (Gross 1975) and stochastic diffusion processes (Bakry-Emery 1983) Toscani (1997): kinetic Fokker-Planck equations
Introduction 3 Example: heat equation t u = u, u(0) = u 0 0 in T d (torus), t > 0 Mass conservation: T d u(t)dx = T d u 0 dx =: ū Lyapunov functional (entropy): H 2 [u] = 1 (u ū) 2 dx 2 T Time derivative of H d 2 : dh 2 = (u ū) t udx = u 2 dx dt T d } T d {{} entropy dissipation Poincaré inequality: u 2 L 2 (T d ) C 1 P u ū 2 L 2 (T d ) Combining expressions: dh 2 dt 0 C P H 2 H 2 [u(t)] H 2 [u 0 ]e C Pt
Introduction 4 Example: heat equation t u = u, u(0) = u 0 in T d (torus), t > 0 Another Lyapunov functional (entropy): H 1 [u] = u log ū T d u dx Time derivative of H 1 : dh 1 = (log ū ) dt T d u + 1 t udx = 4 u 2 dx T d Logarithmic Sobolev inequality: u log ū dx C 1 T d L u 2 dx u T d Combining expressions: dh 1 4C L H 1 H 1 [u(t)] H 1 [u 0 ]e 4C Lt dt
Introduction 5 H 2 [u(t)] H 2 [u 0 ]e C Pt, Example: heat equation H 1 [u(t)] H 1 [u 0 ]e 4C Lt Benefits of entropy-dissipation method: Exponential decay rate to global equilibrium Applicable to nonlinear equations (linear: semigroup theory) Applicable to systems of PDEs: gives maximum principle Relation to functional inequalities Provides proofs of functional ineq. with explicit constants Ingredients of entropy-dissipation method: Entropy functional Entropy-dissipation functional Relation between entropy and entropy dissipation
Introduction 6 Setting: u solves Au = 0, u solves t u + Au = 0, t > 0, u(0) = u 0 Definitions: Lyapunov functional: H satisfies dh dt [u(t)] 0 t Entropy: H is convex Lyapunov functional and φ C 0 (R) with φ(0) = 0 and distance d such that d(u, u ) φ(h[u] H[u ]) for all u Entropy dissipation: D = dh/dt, H being an entropy Example: heat equation, H 2 [u] = 1 2 T d (u u ) 2 dx, φ(x) = x, d(u, v) = u v L 2 d(u, u ) = u u L 2 = H 2 [u] = H 2 [u] H 2 [u ]
Introduction 7 Examples of nonlinear equations: Thin-film equation: t u + div (u β u) = 0 Quantum diffusion equation: ( ) t u + div u u u = 0 Cross-diffusion system: t u div (A( u) u) = R( u), A( u) matrix Examples of entropies: 1 H α [u] = u α dx, α > 0, α 1 α(1 α) Ω H 1 [u] = u(log u 1)dx Ω F α [u] = u α/2 2 dx, α > 0 Ω
Overview 8 Introduction Boltzmann equation Fokker-Planck equations Degenerate parabolic systems
Boltzmann equation 9 f(x, v, t) 0 probability distribution on phase space Elastic collisions: momentum/energy conservation Velocities: (v, w) before collision, (v, w ) after collision v + w = v + w, v 2 + w 2 = v 2 + w 2 Solving system of 2d unknowns with parameter σ S d 1 : v = 1 2 (v + w + v w σ), w = 1 2 (v + w v w σ) Boltzmann equation: t f + v x f = Q(f, f), x, v R d, t > 0 Collision operator: (Boltzmann 1875) Q(f, f)= B( v w, σ)(f(v )f(w ) f(v)f(w))dw dσ S d 1 R n Gain term f(v )f(w ), loss term f(v)f(w) Collision kernel B(z, σ) 0: depends on collision angle
Boltzmann equation 10 Conservation properties t f +v x f = Q(f, f) = B(f(v )f(w ) f(v)f(w))dw dσ Multiply by φ(v) and integrate: d fφ(v)dv + div x fvφ(v)dv= 1 BGdσ dw dv dt 4 G = (f(v )f(w ) f(v)f(w))(φ(v )+φ(w ) φ(v) φ(w)) If φ(v) = 1, v, 1 2 v 2 then G = 0 Conservation of mass/momentum/energy: 1 v d v dv + div x f v v dt R d R d f 1 2 v 2 1 2 v v 2 dv = 0
Boltzmann equation 11 Boltzmann entropy t f +v x f = Q(f, f) = B(f(v )f(w ) f(v)f(w))dw dσ H-theorem: H 1 [f] = R d f log fdv is an entropy satisfying dh 1 [f] = D[f] 0 dt Steady state: f (v) = (2π) d/2 exp( v 2 /2) Relative entropy: H1[f] = H 1 [f] H 1 [f ] Question: Convergence f(t) f? If D[f] λh 1[f] then, by the Gronwall lemma, H 1[f(t)] e λt H 1[f(0)], t > 0 Only (Villani 2003): D[f] λ ε (f)h 1[f] 1+ε for all ε > 0 H 1[f(t)] C ε t 1/ε for all ε > 0
Overview 12 Introduction Boltzmann equation Fokker-Planck equations Degenerate parabolic systems
Fokker-Planck equations 13 Relaxation to self-similarity t u = u in R d, t > 0, u(0) = u 0 0, Self-similar solution: U(x, t) = (2π(2t + 1)) d/2 exp ) ( x 2 2(2t+1) R d u 0 dx = 1 Question: How fast u(t) U(t) 0 as t? Main idea: transformation y = x/ 2t + 1, s = log 2t + 1 leads to Fokker-Planck equation s v = div ( v + yv), v(y,s) = e ds u(e s y, 1 2 (e2s 1)) Solution U(x, t) becomes M(y) = (2π) d/2 exp( y 2 /2) Theorem: Let u 0 L 1 (R d ) be nonnegative with unit mass. Then u(t) U(t) L 1 (2t + 1) 1/2 8H 1 [u 0 ]
Fokker-Planck equations 14 Consider more general setting: t u = div ( u + u V ) in R d, t > 0, u(0) = u 0 Potential: lim x V (x) = Steady state: u = Ze V, Z = ( R d e V dx) 1 Entropy: φ smooth convex, φ(1) ( = ) φ (1) = 0 u H φ [u] = u dx R d φ u Theorem: (Arnold-Markowich-Toscani-Unterreiter 2001) Let u 0 log u 0 L 1, 2 V λ > 0, 1/φ concave. Then H φ [u(t)] e 2λt H φ [u 0 ], t > 0 u(t) u L 1 (R d ) e λt C φ H φ [u 0 ] 1/2 Example: φ(s) = s(log s 1) + 1, V (x) = 1 2 x 2
Fokker-Planck equations 15 Proof: (Bakry/Emery 1983) First time derivative: dh φ = φ (ρ) ρ 2 u dx 0, ρ = u dt R d u Second time derivative: D φ = dh φ dt (entropy dissipation) dd φ = 2λD φ quadratic polynomial in ρ, 2 ρ dt 2λD φ = 2λ dh φ dt Integrate over (t, ): lim D φ[u(s)] } s {{} =0 D φ [u(t)] 2λ lim s H φ [u(s)] }{{} =0 2λH φ [u(t)] Gronwall lemma: dh φ dt 2λH φ H φ [u(t)] e 2λt H φ [u 0 ] Csiszár-Kullback inequality: {}}{ u(t) u L 1 C φ (H φ [u(t)] H φ [u ]) 1/2 C φ e λt H φ [u 0 ] 1/2 =0
Fokker-Planck equations 16 Comments: Difficult part: justify computations for weak solutions Where is the logarithmic Sobolev inequality? Answer: H φ [u]= φ(ρ)u dx 1 R d 2λ Example: V (x) = 1 2 x 2, φ(s) = s(log s 1) + 1 R d u log udx + log(2π) d/2 + d 2 R d φ (ρ) ρ 2 u dx = 1 2λ D φ[u] R d u 2 dx Benefit: Simultaneous proof of decay rate and log Sobolev ineq. Generalization to nonlinear case possible
Fokker-Planck equations 17 Nonlinear Fokker-Planck equation t u = div ( f(u) + u V ) in R 2, t > 0, u(0) = u 0 0 Assume: f(u) = u m with m > 1 (to simplify), 2 V λ > 0 ( ) u H m 1 [u] = H[u] H[u ], H[u] = u R d m 1 + V (x) dx Theorem: (Carrillo-A.J.-Markowich-Toscani-Unterreiter 2001) Let u 0 log u 0 L 1, 2 V λ > 0. Then u(t) u L 1 C(H[u 0 ])e 2λt, t > 0 Proof: First time derivative: D[u] = dh dt 0 Second time derivative: dd dt = 2λD + R(t), R(t) 0 Integrate and use lim t D[u(t)] = 0 (not trivial!) Gronwall lemma and Csiszár-Kullback inequality
Fokker-Planck equations 18 Comments: If V (x) = 1 2 x 2, 1 2 < p = 1 2m 1 < 1, the functional inequality H [u] 1 ( ) u u m 1 2λ R d m 1 + V dx gives Gagliardo-Nirenberg ineq. (Del Pino-Dolbeault 2002) u L p+1 (R d ) C u θ L 2 (R d ) u 1 θ L 2p (R d ), θ = d(1 p) (1+p)(2p+d(1 p)) General nonlinearity: (1 1 d )f(u) uf (u), 2 V λ H [u] C(H [u 0 ])e 2λt, t > 0 Heat equation Poincaré inequality Linear Fokker-Planck log Sobolev inequality Porous-medium Gagliardo-Nirenberg inequality
Overview 19 Introduction Boltzmann equation Fokker-Planck equations Degenerate parabolic systems
Degenerate parabolic systems 20 Aim: Time decay of solutions u(t) R N to u = 0 of t b(u) diva(u, u) = f(u) in R d, t > 0, b(u(0)) = b(u 0 ) Assumptions: b : R N R N monotone, b = χ, b(u) u 1/m a : R N R Nd R elliptic: (a(u, y) a(u, z)) (y z) α y z p, p 2 Examples: Porous-medium equation: b(u) = u 1/m, a = u Energy-transport system: b(u) = (n, 3 2nT), a = A(u) u, where n = T 3/2 e u 1: particle density, T = 1 u 2 : temperature Entropy: Let h(b(u)) = b(u) u χ(u) be the Legendre transform of χ H[u] = h(b(u))dx R d
Degenerate parabolic systems 21 t b(u) diva(u, u) = f(u) in R d, t > 0, b(u(0)) = b(u 0 ) Theorem: (Carrillo-A.J.-Markowich-Toscani-Unterreiter 2001) Let p > dm+d dm+1, γ = dm(p 1)+p d dm. Then u(t) L 1+1/m C 1 (H[u 0 ])(1 + C 2 t) m/(γ(m+1)), m > 1 2 u(t) L 1 C 1 (H[u 0 ])(1 + C 2 t) (m 1)/(γm), m > 1 Proof: Time derivative: dh dt α u p L p Nash inequality: u 1+θ L 1+1/m C u 1/m θm u L 1 L p dh dt α C p u 1/m p/m }{{ L 1 } αh[u] p(1+θ) p(1+θ) u } L{{ 1+1/m } C p M p/m M p/m =H[u] p(1+θ) Apply nonlinear Gronwall lemma and Csiszár-Kullback ineq.
Degenerate parabolic systems 22 t u div (A(u) u + V A(u) u) on T d, u(0) = u 0 Question: Can we extend Bakry-Emery technique to systems? Partial answers: (work in progress with Dolbeault & Matthes) Linear systems: Assume A = const., d = 1 H[u] = u Qudx, Q symm. positive definite T d If AQ 1 symmetric, V xx A λ then H[u(t)] H[u 0 ]e 2λt Nonlinear systems: Assume A(u) = A 0 + εa 1 (u), d = 1 1 N H[u] = u α α(α 1) i=1 T d i dx, α > 1 If A 1 (u) bounded, ε > 0 small, and conditions on V xx, α then κ > 0: H[u(t)] H[u 0 ]e κt
Summary 23 Summary: Entropy/entropy dissipation in kinetic and diffusion eqs. Allows one to prove explicit equilibration rates and convex Sobolev ineq. with (sometimes) optimal constants Extremely flexible for nonlinear equations and systems Discussion: Drawback of Bakry-Emery method: many integrations by parts Can this be made systematic? Can we exploit the entropy structure of diffusion systems? Answer: Yes! (See the next talk)
Summary 24 Thank you!