On <inline-formula><tex-math id="M1">$ C^{2} $</tex-math></inline-formula> solution of the free-transport equation in a disk

Abstract

&lt;p style='text-indent:20px;'&gt;The free transport operator of probability density function &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ f(t, x, v) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is one the most fundamental operator which is widely used in many areas of PDE theory including kinetic theory, in particular. When it comes to general boundary problems in kinetic theory, however, it is well-known that high order regularity is very hard to obtain in general. In this paper, we study the free transport equation in a disk with the specular reflection boundary condition. We obtain initial-boundary compatibility conditions for &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ C^{1}_{t, x, v} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ C^{2}_{t, x, v} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; regularity of the solution. We also provide regularity estimates.&lt;/p&gt;

Hölder regularity of the Boltzmann equation past an obstacle

“…Indeed in [14], they proved this optimal BV regularity for the diffuse and inflow boundary. We also refer very recent result [25] which shows that higher order regularity is hard to obtain in general even for free transport equation in the presence of physical boundary conditions.…”