Global-in-time L p -- L q estimates for solutions of the Kramers-Fokker-Planck equation

TL;DR

This paper establishes optimal global-in-time Lp-Lq estimates for solutions to the 3D Kramers-Fokker-Planck equation with short-range potential, revealing decay and divergence rates comparable to the heat equation and sub-elliptic behavior.

Contribution

It provides the first optimal global-in-time Lp-Lq estimates for the Kramers-Fokker-Planck equation with short-range potential in three dimensions.

Findings

Decay rate matches the heat equation in x-variables as t→∞

Divergence rate as t→0 relates to sub-ellipticity with 1/3 derivative loss

Establishes optimal estimates for solutions in the specified setting

Abstract

In this work, we prove an optimal global-in-time L p --L q estimate for solutions to the Kramers-Fokker-Planck equation with short range potential in dimension three. Our result shows that the decay rate as t $\rightarrow$ +$\infty$ is the same as the heat equation in x-variables and the divergence rate as t $\rightarrow$ 0 + is related to the sub-ellipticity with loss of 1/3 derivatives of the Kramers-Fokker-Planck operator.