Diffusive Limit of the Boltzmann Equation in Bounded Domains

IF 2.2 1区 物理与天体物理 Q1 PHYSICS, MATHEMATICAL
Zhimeng Ouyang, Lei Wu
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引用次数: 0

Abstract

The investigation of rigorous justification of the hydrodynamic limits in bounded domains has seen significant progress in recent years. While some headway has been made for the diffuse-reflection boundary case (Esposito et al. in Ann PDE 4:1–119, 2018; Ghost effect from Boltzmann theory. arXiv:2301.09427, 2023; Jang and Kim in Ann PDE 7:103, 2021), the more intricate in-flow boundary case, where the leading-order boundary layer effect cannot be neglected, still poses an unresolved challenge. In this study, we tackle the stationary and evolutionary Boltzmann equations, considering the in-flow boundary conditions within both convex and non-convex bounded domains, and demonstrate their diffusive limits in \(L^2\). Our approach hinges on a groundbreaking insight: a remarkable gain of \(\varepsilon ^{\frac{1}{2}}\) in the kernel estimate, which arises from a meticulous selection of test functions and the careful application of conservation laws. Additionally, we introduce a boundary layer with a grazing-set cutoff and investigate its BV regularity estimates to effectively control the source terms in the remainder equation with the help of the Hardy’s inequality.

有界域中波尔兹曼方程的扩散极限
近年来,对有界域流体力学极限的严格论证研究取得了重大进展。虽然在扩散-反射边界情况下取得了一些进展(Esposito 等人在 Ann PDE 4:1-119, 2018; Ghost effect from Boltzmann theory. arXiv:2301.09427, 2023; Jang and Kim in Ann PDE 7:103, 2021),但在更复杂的内流边界情况下,前导阶边界层效应不可忽视,这仍然是一个尚未解决的挑战。在本研究中,我们考虑了凸域和非凸有界域中的内流边界条件,处理了静止和演化玻尔兹曼方程,并证明了它们在(L^2\)中的扩散极限。我们的方法依赖于一个突破性的见解:在内核估计中获得了 \(\varepsilon ^{\frac{1}{2}}\)的显著增益,这源于对测试函数的精心选择和对守恒定律的谨慎应用。此外,我们还引入了具有放牧集截止的边界层,并研究了其 BV 正则性估计,以借助哈代不等式有效控制余数方程中的源项。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Communications in Mathematical Physics
Communications in Mathematical Physics 物理-物理:数学物理
CiteScore
4.70
自引率
8.30%
发文量
226
审稿时长
3-6 weeks
期刊介绍: The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.
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