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

2025/11/25 by Hongxu Chen, Renjun Duan, Chen, Hongxu +3
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Spectral Theory in Mathematical Physics #Thermoelastic and Magnetoelastic Phenomena

paper · pdf · doi:10.48550/arxiv.2511.19950

openalex publication_date 2025/11/25 · openalex created_date 2025/11/28 · openalex updated_date 2026/07/28

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 ℝ²× (-1,1). The isothermal diffuse reflection boundary condition is imposed on two parallel infinite planes at x₃=± 1. The main difficulties lie in the fact that the direct Fourier transform is not applicable to the vertical x₃-variable, and the linear collision operator K loses its compactness on L²((-1,1)× \R³_v) although it is compact on L²(\R³_v). By introducing a regularization operator K_n via the finite-dimensional Fourier series truncation in L²(-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→ ∞, 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.

Citations

• Global dynamics of isothermal rarefied gas flows in an infinite layer
• Diffusive Expansion of the Boltzmann equation for the flow past an obstacle
• The 3D kinetic Couette flow via the Boltzmann equation in the diffusive limit
• Global Diffusive Expansion of Boltzmann Equation in exterior Domain
• Gradient Decay in the Boltzmann theory of Non-isothermal boundary
• Passage from the Boltzmann equation with Diffuse Boundary to the Incompressible Euler equation with Heat Convection
• Regularity of Stationary Boltzmann equation in Convex Domains
• Incompressible Euler limit from Boltzmann equation with Diffuse Boundary Condition for Analytic data
• Spectrum Structure and Behaviors of the Vlasov-Maxwell-Boltzmann Systems
• Spectrum analysis for the Vlasov-Poisson-Boltzmann system
• Factorization for non-symmetric operators and exponential H-theorem

Related

• The Boltzmann Equation with Large-amplitude Initial Data in Bounded Domains
• Boundary layers of the Boltzmann equation in a three-dimensional half-space
• The Boltzmann equation with large-amplitude initial data and specular reflection boundary condition
• Global dynamics of isothermal rarefied gas flows in an infinite layer
• The Boltzmann equation in the homogeneous critical regularity framework