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

Hongxu Chen, Renjun Duan, Shuangqian Liu

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.

The Boltzmann equation in an infinite layer: spectrum and asymptotics 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 . The isothermal diffuse reflection boundary condition is imposed on two parallel infinite planes at . The main difficulties lie in the fact that the direct Fourier transform is not applicable to the vertical -variable, and the linear collision operator loses its compactness on although it is compact on . By introducing a regularization operator via the finite-dimensional Fourier series truncation in , 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 , 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.

Paper Structure

This paper contains 13 sections, 11 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Main result
  4. Difficulty and key ideas
  5. Outline
  6. Other notations
  7. Preliminary
  8. Properties of linear operator $K$ and nonlinear operator $\Gamma$
  9. Semi-group $e^{At}$, $e^{Bt}$ and Fourier transform in horizontal direction
  10. Regularization of $K$ and semi-group $e^{B_n t}$
  11. Macroscopic estimate to the eigenvalue problem
  12. Linear estimate
  13. Spectral analysis when $k=0$

Key Result

Theorem 1.1

There exist small constants $\delta>0$ and $c_0>0$ such that if the initial condition $F_0 := \mu + \sqrt{\mu}f_0\geq 0$ satisfies then there exists a unique global solution $F := \mu + \sqrt{\mu}f\geq 0$ to the initial-boundary value problem nonlinear_f for $1\leq q < 2$ such that, If we further assume $\mathbf{P}_0 f_0 \equiv 0$, then the existence and uniqueness of $F=\mu+\sqrt{\mu}f\geq 0$ h

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.2
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1: R
  • Lemma 2.2: G
  • Lemma 2.3: ukai2006boltzmann and G
  • ...and 12 more