Department of Mathematics - Seminar on PDE - Optimal regularity for kinetic equations in domains

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. They established that any possible
singularity of a periodic solution to the Boltzmann equation must be visible
macroscopically. 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. Finally, I will report on recent joint work with Xavier
Ros-Oton, in which we establish sharp boundary regularity results for this class of
equations.