Variational methods for the kinetic Fokker-Planck equation

TL;DR

This paper introduces a functional analytic framework for the kinetic Fokker-Planck equation, establishing well-posedness, regularity, and exponential convergence to equilibrium using new inequalities and energy estimates.

Contribution

It develops a novel $H^1$-type space, proves regularity and convergence results, and introduces new inequalities for the kinetic Fokker-Planck equation.

Findings

Established well-posedness of weak solutions.

Proved $C^ abla$ regularity of solutions.

Demonstrated exponential convergence to equilibrium.

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 Poincar\'e and H\"ormander 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 Poincar\'e-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.