For linear Boltzmann transport problems whose data vanish on the inflow and characteristic boundary parts, strong solutions exist with arbitrary spatial Sobolev regularity, including with a continuous slowing-down energy term.
On the Existence of $H^1$ solutions for Stationary Linearized Boltzmann Equations in a Small Convex Domain
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abstract
In this article, we investigate the incoming boundary value problem for the stationary linearized Boltzmann equations in $ \Omega \subseteq \mathbb{R}^{3}$. For a $C^2$ bounded domain with boundary of positive Gaussian curvature, the existence theory is established in $H^{1}(\Omega \times \mathbb{R}^{3})$ provided that the diameter of the domain $\Omega$ is small enough.
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On existence of spatially regular strong solutions for a class of transport equations
For linear Boltzmann transport problems whose data vanish on the inflow and characteristic boundary parts, strong solutions exist with arbitrary spatial Sobolev regularity, including with a continuous slowing-down energy term.