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arXiv · 2502.03096

BGK model for rarefied gas in a bounded domain

Hongxu Chen·Christian Klingenberg·Marlies Pirner
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Abstract

We study the Bathnagar-Gross-Krook (BGK) equation in a smooth bounded domain featuring a diffusive reflection boundary condition with general collision frequency. We prove that the BGK equation admits a unique global solution with an exponential convergence rate if the initial condition is a small perturbation around the global Maxwellian in the $L^\infty$ space. For the proof, we utilize the dissipative nature from the linearized BGK operator and establish an $L^2$ coercive estimate. Next, we derive the a priori estimate by obtaining an $L^\infty$ bound on the nonlinear operator; this requires a delicate analysis to manage its intrinsic nonlinear structure. Finally, we establish the $L^\infty$ stability estimate and introduce sequential arguments for the nonlinear BGK operator, thereby concluding both well-posedness and positivity.

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Hongxu Chen, Christian Klingenberg, Marlies Pirner. 2025-02-05. BGK model for rarefied gas in a bounded domain. https://arxiv.org/abs/2502.03096

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