Kinetic Fokker-Planck and Landau equations with specular reflection boundary condition

Abstract

<p style='text-indent:20px;'>We establish existence of finite energy weak solutions to the kinetic Fokker-Planck equation and the linear Landau equation near Maxwellian, in the presence of specular reflection boundary condition for general domains. Moreover, by using a method of reflection and the <inline-formula><tex-math id="M1">\begin{document}$ S_p $\end{document}</tex-math></inline-formula> estimate of [<xref ref-type="bibr" rid="b7">7</xref>], we prove regularity in the kinetic Sobolev spaces <inline-formula><tex-math id="M2">\begin{document}$ S_p $\end{document}</tex-math></inline-formula> and anisotropic Hölder spaces for such weak solutions. Such <inline-formula><tex-math id="M3">\begin{document}$ S_p $\end{document}</tex-math></inline-formula> regularity leads to the uniqueness of weak solutions.</p>