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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.
}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2021-0081}, url = {http://global-sci.org/intro/article_detail/cmr/21072.html} }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.