The Boltzmann equation in an infinite layer: spectrum and asymptotics toward the heat equation

TL;DR

This paper develops spectral theory to analyze the asymptotic behavior of solutions to the Boltzmann equation in an infinite layer, showing convergence to a heat equation in large-time limits.

Contribution

It introduces a regularization approach to study the spectrum of the linearized Boltzmann operator in a bounded domain, linking spectral properties to asymptotic dynamics.

Findings

Established resolvent estimates for the regularized operator

Identified the leading diffusive eigenvalue governing asymptotics

Proved convergence rate of solutions toward the heat equation

Abstract

In the paper, we develop spectral theory to analyze the sharp asymptotic behavior of solutions to the Boltzmann equation around global Maxwellians in a three-dimensional infinite layer $\mathbb{R}^2\times (-1,1)$. The isothermal diffuse reflection boundary condition is imposed on two parallel infinite planes at $x_3=\pm 1$. The main difficulties lie in the fact that the direct Fourier transform is not applicable to the vertical $x_3$-variable, and the linear collision operator $K$ loses its compactness on $L^2((-1,1)\times \R^3_v)$ although it is compact on $L^2(\R^3_v)$. By introducing a regularization operator $K_n$ via the finite-dimensional Fourier series truncation in $L^2(-1,1)$, we study the spectrum of the linearized initial-boundary value approximation problem, establish the resolvent estimates, and identify the leading diffusive eigenvalue. This spectral structure governs the sharp asymptotic dynamics of the original linear problem as $n\to \infty$, enabling us to construct the large-time behavior for the nonlinear problem and rigorously prove that the solution converges with a faster rate toward that of the two-dimensional heat equation in the horizontal direction.