Variational methods for the kineticFokker–Planck equation

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-and Hrmander-type and combine them with basic energy estimates (analogous to the Caccioppoli inequality) in an iteration procedure to obtain the C regularity of weak solutions.We also use the Poincar-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.which is often called the kinetic Fokker-Planck equation.