Variational methods for the kinetic Fokker-Planck equation (Q6596280)

In this paper, the authors 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. Specifically, they introduce 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, they identify the weak solution as the minimizer of a uniformly convex functional. By making use of new functional inequalities of Poincaré- and Hörmander-type together with basic energy estimates, they established the \(C^{\infty}\) regularity of weak solutions. They 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 classical dissipative estimate for the heat equation. Moreover, they prove enhanced dissipation in a weakly collisional limit.