Asymptotic Analysis of Boltzmann Equation in Bounded Domains

Lei Wu; Zhimeng Ouyang

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arXiv · 2008.10507

Asymptotic Analysis of Boltzmann Equation in Bounded Domains

Lei Wu·Zhimeng Ouyang

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Abstract

Consider 3D Boltzmann equation in convex domains with diffusive-reflection boundary condition. We study the hydrodynamic limits as the Knudsen number and Strouhal number $\epsilon\rightarrow 0^+$. Using the Hilbert expansion, we rigorously justify that the solution of stationary/evolutionary problem converges to that of the steady/unsteady Navier-Stokes-Fourier system. This is the first paper to justify the hydrodynamic limits of nonlinear Boltzmann equations with hard-sphere collision kernel in bounded domain in the $L^{\infty}$ sense. The proof relies on a novel analysis on the boundary layer effect with geometric correction. The difficulty mainly comes from three sources: 3D domain, boundary layer regularity, and time dependence. To fully solve this problem, we introduce several techniques: (1) boundary layer with geometric correction; (2) remainder estimates with $L^2-L^{6}-L^{\infty}$ framework. Keywords: boundary layer; Milne problem; geometric correction; remainder estimates.

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Lei Wu, Zhimeng Ouyang. 2020-08-24. Asymptotic Analysis of Boltzmann Equation in Bounded Domains. https://arxiv.org/abs/2008.10507

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