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arXiv · 2311.12387

Geometric effects on $W^{1, p}$ regularity of the stationary linearized Boltzmann equation

I-Kun Chen·Chun-Hsiung Hsia·Daisuke Kawagoe·Jhe-Kuan Su
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Abstract

We study the incoming boundary value problem for the stationary linearized Boltzmann equation in bounded convex domains. The geometry of the domain has a dramatic effect on the space of solutions. We prove the existence of solutions in $W^{1,p}$ spaces for $1 \leq p<2$ for small domains. In contrast, if we further assume the positivity of the Gaussian curvature on the boundary, we prove the existence of solutions in $W^{1, p}$ spaces for $1 \leq p < 3$ provided that the diameter of the domain is small enough. In both cases, we provide counterexamples in the hard sphere model; a bounded convex domain with a flat boundary for $p = 2$, and a small ball for $p = 3$.

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I-Kun Chen, Chun-Hsiung Hsia, Daisuke Kawagoe, Jhe-Kuan Su. 2023-11-21. Geometric effects on $W^{1, p}$ regularity of the stationary linearized Boltzmann equation. https://doi.org/10.1512/iumj.2025.74.60550

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