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

2025/07/24 by Daisuke Kawagoe, Kawagoe, Daisuke
Computer Science · Mathematics · #35F30 #35Q20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2507.18211

openalex publication_date 2025/07/24 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28

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 ≤ γ≤ 1), while H^{((4 + γ)/2)-}_x regularity is obtained for very soft potentials (-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 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.

Related

• A Revisit of the Velocity Averaging Lemma: on the Regularity of Stationary Boltzmann Equation in a Bounded Convex Domain
• Regularity of Stationary Boltzmann equation in Convex Domains
• Regularity of stationary solutions to the linearized Boltzmann equations
• On regularity of a Kinetic Boundary layer
• Geometric effects on W^{1, p} regularity of the stationary linearized Boltzmann equation