Variational methods for the kinetic Fokker-Planck equation

Understand

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.