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Variational methods for the kinetic Fokker-Planck equation

D. Albritton, S. Armstrong, J. -C. Mourrat, M. Novack

Abstract

We develop a functional analytic approach to the study of the Kramers and kinetic Fokker-Planck equations which parallels the classical $H^1$ theory of uniformly elliptic equations. In particular, we identify a function space analogous to $H^1$ and develop a well-posedness theory for weak solutions in this space. In the case of a conservative force, we identify the weak solution as the minimizer of a uniformly convex functional. We prove new functional inequalities of Poincaré and Hörmander type and combine them with basic energy estimates (analogous to the Caccioppoli inequality) in an iteration procedure to obtain the $C^\infty$ regularity of weak solutions. We also use the Poincaré-type inequality to give an elementary proof of the exponential convergence to equilibrium for solutions of the kinetic Fokker-Planck equation which mirrors the classic dissipative estimate for the heat equation. Finally, we prove enhanced dissipation in a weakly collisional limit.

Variational methods for the kinetic Fokker-Planck equation

Abstract

We develop a functional analytic approach to the study of the Kramers and kinetic Fokker-Planck equations which parallels the classical theory of uniformly elliptic equations. In particular, we identify a function space analogous to and develop a well-posedness theory for weak solutions in this space. In the case of a conservative force, we identify the weak solution as the minimizer of a uniformly convex functional. We prove new functional inequalities of Poincaré and Hörmander type and combine them with basic energy estimates (analogous to the Caccioppoli inequality) in an iteration procedure to obtain the regularity of weak solutions. We also use the Poincaré-type inequality to give an elementary proof of the exponential convergence to equilibrium for solutions of the kinetic Fokker-Planck equation which mirrors the classic dissipative estimate for the heat equation. Finally, we prove enhanced dissipation in a weakly collisional limit.

Paper Structure

This paper contains 25 sections, 30 theorems, 353 equations.

Table of Contents

  1. Introduction
  2. Motivation and informal summary of results
  3. Statements of the main results
  4. On unique solvability of the Dirichlet problem
  5. Outline of the paper
  6. Function space basics
  7. Properties of honegamma and hminusonegamma
  8. Density of Smooth Functions in H1hyyp
  9. Besov Spaces
  10. Functional inequalities for H1hyp
  11. The Poincaré inequality for H1hyp
  12. Poincaré inequality with confining potential
  13. Interpolation and Hörmander inequalities for H1hyp
  14. Compact embedding of H1hyp into L2
  15. The Kramers equation
  16. ...and 10 more sections

Key Result

Theorem 1.2

Let $\mathbf{b}$ satisfy Assumption a.conserv, and let $f^*\in L^2(\mathbb{T}^d;H^{-1}_\gamma)$ be such that $\iint_{\mathbb{T}^d\times\mathbb{R}^d} f^*(x,v) \, dm = 0$. Then there exists a unique weak solution $f \in H^1_{\mathrm{hyp}}(\mathbb{T}^d)$ to the Kramers equation with $\iint_{\mathbb{T}^d\times\mathbb{R}^d} f(x,v) \,dm = 0$. Furthermore, there exists a constant $C(\mathbf{b},d)<\inft

Theorems & Definitions (64)

  • Theorem 1.2: Well-posedness of the Kramers equation
  • Theorem 1.3: Poincaré inequality for $H^1_{\mathrm{hyp}}$
  • Theorem 1.4: Hörmander inequality for $H^1_{\mathrm{hyp}}$
  • Theorem 1.5: Interior Sobolev regularity for \ref{['e.thepde']}
  • Theorem 1.6: Convergence to equilibrium
  • Theorem 1.7: Enhanced dissipation
  • Lemma 2.1: Identification of $H^{-1}_\gamma$
  • proof
  • Proposition 2.2
  • proof
  • ...and 54 more