论文标题
界限中传输方程的渐近分析
Asymptotic Analysis of Transport Equation in Bounded Domains
论文作者
论文摘要
考虑具有流量内边界的3D凸域中的中子传输方程。我们主要研究渐近极限，因为Knudsen Number $ε\ rightarrow 0^+$。使用希尔伯特的扩展，我们严格地证明，稳定问题的解决方案会收敛到拉普拉斯方程的解决方案，而不稳定问题的解决方案会收敛到热方程。证明依赖于对边界层效应进行几何校正的详细分析。这个问题可以在许多不同的环境中提出，上述问题可能是最重要的，数学上最具挑战性的。在过去的十年中，我们必须利用我们在一系列论文中开发的几乎所有方法和技术，并带来新颖的想法来治疗新的并发症。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^{2m} -l^{\ infty} $ framework;边界层分解。
Consider neutron transport equations in 3D convex domains with in-flow boundary. We mainly study the asymptotic limits as the Knudsen number $ε\rightarrow 0^+$. Using Hilbert expansion, we rigorously justify that the solution of steady problem converges to that of the Laplace's equation, and the solution of unsteady problem converges to that of the heat equation. The proof relies on a detailed analysis on the boundary layer effect with geometric correction. This problem can be formulated in many different settings, and the above one is probably the most physically significant and most mathematically challenging. We have to utilize almost all methods and techniques we developed in a series of papers in the past decade, and bring novel ideas to treat the new complications. 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 $L^2-L^{2m}-L^{\infty}$ framework; boundary layer decomposition.
