$H^s_x$ regularity of solutions to the stationary Boltzmann equation with the incoming boundary condition

TL;DR

This paper investigates the fractional Sobolev regularity of solutions to the stationary Boltzmann equation with incoming boundary conditions, revealing new regularity results for different potential types without assuming boundary curvature positivity.

Contribution

It establishes $H^{s}_x$ regularity results for solutions to the stationary Boltzmann equation under less restrictive boundary conditions and potential assumptions, extending previous regularity theories.

Findings

Solutions have $H^{1-}_x$ regularity for hard and soft potentials.

For very soft potentials, solutions have $H^{((4 + ext{ extgamma})/2)-}_x$ regularity.

The paper develops new $L^2-L^$ estimates and extends regularity results to nonlinear problems.

Abstract

We consider the stationary Boltzmann equation with the angular cutoff cross section in a bounded convex domain under the incoming boundary condition. In this article, we discuss the fractional Sobolev regularity of the solution without assuming the positivity of the Gaussian curvature on the boundary. For a boundary data sufficiently smooth and close to the standard Maxwellian, the solution has $H^{1-}_x$ regularity for hard potentials and soft potentials ($-2 \leq \gamma \leq 1$), while $H^{((4 + \gamma)/2)-}_x$ regularity is obtained for very soft potentials ($-3 < \gamma < -2$). We first show the well-posedness of the linearized problem on a weighted $L^2$ space and develop the $L^2-L^\infty$ estimate without the stochastic cycle. We next investigate $H^s_x$ regularity of the solution to the linearized problem. The idea of the celebrated velocity averaging lemma plays a key role in our analysis. We finally derive a bilinear estimate to extend the result on the linearized problem to the weakly nonlinear problem.