Hˢₓ regularity of solutions to the stationary Boltzmann equation with the incoming boundary condition

We consider the stationary Boltzmann equation with the cross section of the form B(|v - ṽ, θ|) = B₀ |v - ṽ|^γcos θsin θ for -3 < γ≤ 1 in a bounded convex domain under the incoming boundary condition. In this article, we shall show the existence of a solution in a weighted L^∞ space with fractional Sobolev regularity without assuming the positivity of the Gaussian curvature on the boundary. For boundary data sufficiently smooth and close to the standard Maxwellian, the solution has H¹⁻ₓ regularity for -2 ≤ γ≤ 1, while only worse regularity is obtained for $-3 < γ< -2$. We first show the well-posedness of the linearized problem on a weighted L² space and develop the L²-L^∞ estimate without the stochastic cycle. We next investigate Hˢₓ regularity of the solution to the linearized problem. The velocity averaging lemma plays a key role in our analysis. We finally derive a bilinear estimate to extend results on the linearized problem to the weakly nonlinear problem.