|
|||||
 |
|
|
|
|
|
|
Variational methods
for the kinetic Fokker–Planck equation
Dallas Albritton, Scott Armstrong, Jean-Christophe Mourrat and Matthew Novack |
|
Vol. 17 (2024), No. 6, 1953–2010
|
 |
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. |
Keywords
kinetic Fokker–Planck equation, hypoelliptic equation,
hypoelliptic diffusion, Poincaré inequality, convergence to
equilibrium, enhancement
|
Mathematical Subject Classification
Primary: 35H10
Secondary: 35D30, 35K70
|
Milestones
Received: 3 November 2021
Revised: 19 February 2023
Accepted: 31 March 2023
Published: 19 July 2024
|
Authors |
| Dallas Albritton | |
| Department of Mathematics University of Wisconsin Madison, WI United States |
|
| Scott Armstrong | |
| Courant Institute of Mathematical
Sciences New York University New York, NY United States |
|
| Jean-Christophe Mourrat | |
| ENS Lyon, CNRS Lyon France |
Courant Institute of Mathematical
Sciences New York University New York, NY United States |
| Matthew Novack | |
| Department of Mathematics Purdue University West Lafayette, IN United States |
|
| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
Open Access made possible by participating institutions via Subscribe to Open.
