The Landau and non-cutoff Boltzmann equations in union of cubes

Abstract

The existence and stability of collisional kinetic equations, especially non-cutoff Boltzmann equation, in a bounded domain with physical boundary condition is a longstanding open problem. This work proves the global stability of the Landau equation and non-cutoff Boltzmann equation in union of cubes with the specular reflection boundary condition when an initial datum is near Maxwellian. Moreover, the solution enjoys exponential large-time decay in the union of cubes. Our method is based on the fact that normal derivatives in cubes are also derivatives along the axis, which allows us to obtain high-order derivative estimates.