Diffusive Limit of the Boltzmann Equation in Bounded Domains

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.