论文标题
界限域中玻尔兹曼方程的渐近分析
Asymptotic Analysis of Boltzmann Equation in Bounded Domains
论文作者
论文摘要
考虑具有扩散反射边界条件的凸域中的3D玻尔兹曼方程。我们将流体动力限制作为Knudsen编号和曲线数$ε\ rightarrow 0^+$。使用希尔伯特的扩展，我们严格地证明，固定/进化问题的解决方案会收敛到稳定/不稳定的Navier-Stokes-tokes-foury System的解决方案。 这是第一篇论文，证明在$ l^{\ infty} $ sense中，具有硬球碰撞内核的非线性玻尔兹曼方程的流体动力限制是合理的。证明依靠对边界层效应的新分析和几何校正。 困难主要来自三个来源：3D域，边界层规律性和时间依赖性。为了充分解决此问题，我们引入了几种技术：（1）边界层具有几何校正； （2）剩余的估计值，$ l^2-l^{6} -l^{\ infty} $ framework。 关键字：边界层；米尔恩问题；几何校正；剩余的估计。
Consider 3D Boltzmann equation in convex domains with diffusive-reflection boundary condition. We study the hydrodynamic limits as the Knudsen number and Strouhal number $ε\rightarrow 0^+$. 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.
