L 2 Hypocoercivity Jean Dolbeault http://www.ceremade.dauphine.fr/ ∼ dolbeaul Ceremade, Université Paris-Dauphine October 16, 2019 CIRM conference on PDE/Probability Interactions Particle Systems, Hyperbolic Conservation Laws October 14-18, 2019
Abstract method and motivation The compact case The non-compact case Vlasov-Poisson-Fokker-Planck Outline Abstract method and motivation ⊲ Abstract statement in a Hilbert space ⊲ Diffusion limit, toy model The compact case ⊲ Strong confinement ⊲ Mode-by-mode decomposition ⊲ Application to the torus ⊲ Further results The non-compact case ⊲ Without confinement: Nash inequality ⊲ With very weak confinement: Caffarelli-Kohn-Nirenberg inequality ⊲ With sub-exponential equilibria: weighted Poincaré inequality The Vlasov-Poisson-Fokker-Planck system ⊲ Linearized system and hypocoercivity ⊲ Results in the diffusion limit and in the non-linear case L2 Hypocoercivity J. Dolbeault
Abstract method and motivation An abstract hypocoercivity result The compact case Diffusion limit The non-compact case Toy model Vlasov-Poisson-Fokker-Planck Abstract method and motivation ⊲ Abstract statement ⊲ Diffusion limit ⊲ A toy model Collaboration with C. Mouhot and C. Schmeiser L2 Hypocoercivity J. Dolbeault
Abstract method and motivation An abstract hypocoercivity result The compact case Diffusion limit The non-compact case Toy model Vlasov-Poisson-Fokker-Planck An abstract evolution equation Let us consider the equation dF dt + T F = L F In the framework of kinetic equations, T and L are respectively the transport and the collision operators We assume that T and L are respectively anti-Hermitian and Hermitian operators defined on the complex Hilbert space ( H , �· , ·� ) � � − 1 ( T Π) ∗ 1 + ( T Π) ∗ T Π A := ∗ denotes the adjoint with respect to �· , ·� Π is the orthogonal projection onto the null space of L L2 Hypocoercivity J. Dolbeault
Abstract method and motivation An abstract hypocoercivity result The compact case Diffusion limit The non-compact case Toy model Vlasov-Poisson-Fokker-Planck The assumptions λ m , λ M , and C M are positive constants such that, for any F ∈ H ⊲ microscopic coercivity: − � L F, F � ≥ λ m � (1 − Π) F � 2 (H1) ⊲ macroscopic coercivity: � T Π F � 2 ≥ λ M � Π F � 2 (H2) ⊲ parabolic macroscopic dynamics: Π T Π F = 0 (H3) ⊲ bounded auxiliary operators: � AT (1 − Π) F � + � AL F � ≤ C M � (1 − Π) F � (H4) The estimate 1 d dt � F � 2 = � L F, F � ≤ − λ m � (1 − Π) F � 2 2 is not enough to conclude that � F ( t, · ) � 2 decays exponentially L2 Hypocoercivity J. Dolbeault
Abstract method and motivation An abstract hypocoercivity result The compact case Diffusion limit The non-compact case Toy model Vlasov-Poisson-Fokker-Planck Equivalence and entropy decay For some δ > 0 to be determined later, the L 2 entropy / Lyapunov functional is defined by 2 � F � 2 + δ Re � A F, F � H [ F ] := 1 so that � AT Π F, F � ∼ � Π F � 2 and − d dt H [ F ] = : D [ F ] = − � L F, F � + δ � AT Π F, F � − δ Re � TA F, F � + δ Re � AT (1 − Π) F, F � − δ Re � AL F, F � ⊲ entropy decay rate: for any δ > 0 small enough and λ = λ ( δ ) λ H [ F ] ≤ D [ F ] ⊲ norm equivalence of H [ F ] and � F � 2 2 − δ � F � 2 ≤ H [ F ] ≤ 2 + δ � F � 2 4 4 L2 Hypocoercivity J. Dolbeault
Abstract method and motivation An abstract hypocoercivity result The compact case Diffusion limit The non-compact case Toy model Vlasov-Poisson-Fokker-Planck Exponential decay of the entropy � � � � λ M λ m λ M , δ = 1 λ m λ M λ = 3 (1+ λ M ) min 1 , λ m , 2 min 1 , λ m , (1+ λ M ) C 2 (1+ λ M ) C 2 M M � � � δ λ M � λ m − δ − 2 + δ − 2 + δ h 1 ( δ, λ ) := ( δ C M ) 2 − 4 λ λ 4 1 + λ M 4 Theorem Let L and T be closed linear operators (respectively Hermitian and anti-Hermitian) on H . Under (H1) – (H4) , for any t ≥ 0 H [ F ( t, · )] ≤ H [ F 0 ] e − λ ⋆ t where λ ⋆ is characterized by � � λ > 0 : ∃ δ > 0 s.t. h 1 ( δ, λ ) = 0 , λ m − δ − 1 λ ⋆ := sup 4 (2 + δ ) λ > 0 L2 Hypocoercivity J. Dolbeault
Abstract method and motivation An abstract hypocoercivity result The compact case Diffusion limit The non-compact case Toy model Vlasov-Poisson-Fokker-Planck Sketch of the proof � � − 1 ( T Π) ∗ T Π, from (H1) and (H2) 1 + ( T Π) ∗ T Π Since AT Π = δ λ M − � L F, F � + δ � AT Π F, F � ≥ λ m � (1 − Π) F � 2 + � Π F � 2 1 + λ M By (H4), we know that | Re � AT (1 − Π) F, F � + Re � AL F, F �| ≤ C M � Π F � � (1 − Π) F � The equation G = A F is equivalent to ( T Π) ∗ F = G + ( T Π) ∗ T Π G � TA F, F � = � G, ( T Π) ∗ F � = � G � 2 + � T Π G � 2 = � A F � 2 + � TA F � 2 � G, ( T Π) ∗ F � ≤ � TA F � � (1 − Π) F � ≤ 1 2 µ � TA F � 2 + µ 2 � (1 − Π) F � 2 � A F � ≤ 1 2 � (1 − Π) F � , � TA F � ≤ � (1 − Π) F � , |� TA F, F �| ≤ � (1 − Π) F � 2 With X := � (1 − Π) F � and Y := � Π F � D [ F ] − λ H [ F ] ≥ ( λ m − δ ) X 2 + δ λ M Y 2 − δ C M X Y − 2 + δ λ ( X 2 + Y 2 ) 1 + λ M 4 L2 Hypocoercivity J. Dolbeault
Abstract method and motivation An abstract hypocoercivity result The compact case Diffusion limit The non-compact case Toy model Vlasov-Poisson-Fokker-Planck Hypocoercivity Corollary For any δ ∈ (0 , 2) , if λ ( δ ) is the largest positive root of h 1 ( δ, λ ) = 0 for which λ m − δ − 1 4 (2 + δ ) λ > 0 , then for any solution F of the evolution equation � F ( t ) � 2 ≤ 2 + δ 2 − δ e − λ ( δ ) t � F (0) � 2 ∀ t ≥ 0 From the norm equivalence of H [ F ] and � F � 2 2 − δ � F � 2 ≤ H [ F ] ≤ 2 + δ � F � 2 4 4 � F 0 � 2 ≤ H [ F 0 ] so that λ ⋆ ≥ sup δ ∈ (0 , 2) λ ( δ ) We use 2 − δ 4 L2 Hypocoercivity J. Dolbeault
Abstract method and motivation An abstract hypocoercivity result The compact case Diffusion limit The non-compact case Toy model Vlasov-Poisson-Fokker-Planck Formal macroscopic (diffusion) limit Scaled evolution equation ε dF dt + T F = 1 ε L F on the Hilbert space H . F ε = F 0 + ε F 1 + ε 2 F 2 + O ( ε 3 ) as ε → 0 + ε − 1 : L F 0 = 0 , ε 0 : T F 0 = L F 1 , ε 1 : dF 0 dt + T F 1 = L F 2 The first equation reads as u = F 0 = Π F 0 The second equation is simply solved by F 1 = − ( T Π) F 0 After projection, the third equation is d dt (Π F 0 ) − Π T ( T Π) F 0 = Π L F 2 = 0 ∂ t u + ( T Π) ∗ ( T Π) u = 0 dt � u � 2 = − 2 � ( T Π) u � 2 ≤ − 2 λ M � u � 2 d is such that L2 Hypocoercivity J. Dolbeault
Abstract method and motivation An abstract hypocoercivity result The compact case Diffusion limit The non-compact case Toy model Vlasov-Poisson-Fokker-Planck A toy problem � � � � du 0 0 0 − k k 2 ≥ Λ > 0 dt = ( L − T ) u , L = , T = , 0 − 1 k 0 Non-monotone decay, a well known picture: see for instance (Filbet, Mouhot, Pareschi, 2006) � � dt | u | 2 = d d u 2 1 + u 2 = − 2 u 2 H-theorem: 2 2 dt � du 1 � dt = − k 2 u 1 macroscopic/diffusion limit: generalized entropy: H ( u ) = | u | 2 − δ k 1+ k 2 u 1 u 2 � � δ k 2 δ k 2 d H δ k u 2 1 + k 2 u 2 = − 2 − 2 − 1 + 1 + k 2 u 1 u 2 1 + k 2 dt δ Λ 1 + δ − (2 − δ ) u 2 1 + Λ u 2 ≤ 2 − 2 u 1 u 2 L2 Hypocoercivity J. Dolbeault
Abstract method and motivation An abstract hypocoercivity result The compact case Diffusion limit The non-compact case Toy model Vlasov-Poisson-Fokker-Planck Plots for the toy problem u1 2 u1 2 , u1 2 + u2 2 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 1 2 3 4 5 6 1 2 3 4 5 6 H D 1.0 1.2 0.8 1.0 0.8 0.6 0.6 0.4 0.4 0.2 0.2 1 2 3 4 5 6 1 2 3 4 5 6 L2 Hypocoercivity J. Dolbeault
Abstract method and motivation An abstract hypocoercivity result The compact case Diffusion limit The non-compact case Toy model Vlasov-Poisson-Fokker-Planck Some references C. Mouhot and L. Neumann. Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity, 19(4):969-998, 2006 F. Hérau. Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation. Asymptot. Anal., 46(3-4):349-359, 2006 J. Dolbeault, P. Markowich, D. Oelz, and C. Schmeiser. Non linear diffusions as limit of kinetic equations with relaxation collision kernels. Arch. Ration. Mech. Anal., 186(1):133-158, 2007. J. Dolbeault, C. Mouhot, and C. Schmeiser. Hypocoercivity for kinetic equations with linear relaxation terms. Comptes Rendus Mathématique, 347(9-10):511 - 516, 2009 J. Dolbeault, C. Mouhot, and C. Schmeiser. Hypocoercivity for linear kinetic equations conserving mass. Transactions of the American Mathematical Society, 367(6):3807-3828, 2015 L2 Hypocoercivity J. Dolbeault
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