| Abstract: |
| The Boltzmann equation is one of the central equations in statistical mechanics and models the evolution of a gas through particle interactions. In recent years, groundbreaking work by Imbert and Silvestre has led to a conditional regularity theory for periodic solutions of the Boltzmann equation. A major open challenge is whether such a theory can be extended to bounded domains with physically relevant boundary conditions. As a first step toward understanding the boundary case, in this talk I will discuss the smoothness of solutions to linear kinetic Fokker-Planck equations in domains with specular reflection condition. While the interior regularity of such equations is well understood, their behavior near the boundary has remained open, even in the simplest case of Kolmogorov`s equation. I will also mention recent results on other boundary conditions such as diffuse reflection and in-flow. This talk is based on joint works with Xavier Ros-Oton and Kyeongbae Kim. |
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