Global-in-Time $L^p−L^q$ Estimates for Solutions of the Kramers-Fokker-Planck Equation

Global-in-Time $L^p−L^q$ Estimates for Solutions of the Kramers-Fokker-Planck Equation

Year:    2022

Author:    Xue-Ping Wang, Lu Zhu

Communications in Mathematical Research , Vol. 38 (2022), Iss. 4 : pp. 560–578

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 $t→ +∞$ is the same as the heat equation in $x$-variables and the divergence rate as $t→0_+$ is related to the sub-ellipticity with loss of one third derivatives of the Kramers-Fokker-Planck operator.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cmr.2021-0081

Communications in Mathematical Research , Vol. 38 (2022), Iss. 4 : pp. 560–578

Published online:    2022-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    19

Keywords:    Global-in-time estimates non-self-adjoint operators kinetic equation Kramers-Fokker-Planck operator.

Author Details

Xue-Ping Wang

Lu Zhu