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A note on asymptotics of linear dissipative kinetic equations in bounded domains

Yuzhe Zhu

TL;DR

The paper analyzes linear dissipative kinetic equations in bounded domains under general Maxwell-type boundary conditions, establishing $L^2$-exponential decay uniform in the Knudsen parameter $\\varepsilon$ and deriving $L^2$ diffusion limits as $\\varepsilon\to0$. It combines energy estimates with $L^2$-hypocoercivity and relative-entropy methods, and introduces a macro-micro decomposition plus a modified entropy to handle non-conservative boundary interactions. The diffusion limit is captured by a parabolic equation for the macroscopic density with Neumann boundary conditions, and the authors provide convergence rates under both well-prepared and general initial data, including initial-layer corrections. The results extend known hypocoercivity and diffusion-limit analyses to bounded domains with grazing boundaries and non-mass-conserving boundary conditions, offering explicit quantitative decay and convergence rates.

Abstract

We establish $L^2$-exponential decay properties for linear dissipative kinetic equations, including the time-relaxation and Fokker-Planck models, in bounded spatial domains with general boundary conditions that may not conserve mass. Their diffusion asymptotics in $L^2$ is also derived under general Maxwell boundary conditions. The proofs are simply based on energy estimates together with previous ideas from $L^2$-hypocoercivity and relative entropy methods.

A note on asymptotics of linear dissipative kinetic equations in bounded domains

TL;DR

The paper analyzes linear dissipative kinetic equations in bounded domains under general Maxwell-type boundary conditions, establishing -exponential decay uniform in the Knudsen parameter and deriving diffusion limits as . It combines energy estimates with -hypocoercivity and relative-entropy methods, and introduces a macro-micro decomposition plus a modified entropy to handle non-conservative boundary interactions. The diffusion limit is captured by a parabolic equation for the macroscopic density with Neumann boundary conditions, and the authors provide convergence rates under both well-prepared and general initial data, including initial-layer corrections. The results extend known hypocoercivity and diffusion-limit analyses to bounded domains with grazing boundaries and non-mass-conserving boundary conditions, offering explicit quantitative decay and convergence rates.

Abstract

We establish -exponential decay properties for linear dissipative kinetic equations, including the time-relaxation and Fokker-Planck models, in bounded spatial domains with general boundary conditions that may not conserve mass. Their diffusion asymptotics in is also derived under general Maxwell boundary conditions. The proofs are simply based on energy estimates together with previous ideas from -hypocoercivity and relative entropy methods.
Paper Structure (8 sections, 8 theorems, 130 equations)

This paper contains 8 sections, 8 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Boundary estimates
  5. Dissipation estimates
  6. Long-time asymptotics
  7. Diffusion asymptotics
  8. Elliptic estimates under the Robin boundary condition

Key Result

Theorem 1.1

Let $\varepsilon\in(0,1]$, $f_{\rm in}\in L^2(\Omega\times\mathbb{R}^d,\mathop{}\!\mathrm{d} m)$, $M_0:=\left(\int_\Omega e^{-\phi}\mathop{}\!\mathrm{d} x\right)^{\!-1} \int_{\Omega\times\mathbb{R}^d} f_{\rm in}\mathop{}\!\mathrm{d} m$, and the functions $\alpha,\beta:\partial\Omega\rightarrow[0,1]$ If $\alpha+\beta\in[0,\delta]$ for some constant $\delta\in[0,1)$, then there are some constants $C

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 7 more