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The Boltzmann equation on smooth and cylindrical domains with Maxwell boundary conditions

Richard Medina Rodriguez

TL;DR

This work studies the Boltzmann equation near its hydrodynamic limit on bounded domains with Maxwell boundary conditions, addressing both smooth $C^2$ domains and cylindrical geometries with a mix of diffusive and specular reflections. It develops a constructive, quantitative $L^2$-$L^\infty$ theory by combining hypocoercivity with the stretching method, and handles both strongly and weakly confining weights, including spatially varying and discontinuous accommodation coefficients. The authors prove well-posedness and exponential relaxation to equilibrium in weighted $L^\infty$ spaces for small $\varepsilon$, using a rescaled formulation, a detailed regularity analysis in cylinders (Poisson and Lamé systems), and a careful operator splitting for weak confinement. The results extend prior cylindrical-domain analyses to variable accommodation, provide explicit decay rates with constructive constants, and offer a robust framework for the Boltzmann equation near the hydrodynamic limit in nontrivial geometries. This has potential implications for quantitative hydrodynamic limits and kinetic-fluid coupling in bounded domains with realistic boundary interactions.

Abstract

In this article we study the well-posedness of the Boltzmann equation near its hydrodynamic limit on a bounded domain. We consider two types of domains, namely $C^2$ domains with Maxwell boundary conditions where the accommodation coefficient is a continuous space dependent function $ι\in [ι_0,1]$ for any $ι_0 \in (0,1]$, or cylindrical domains with diffusive reflection on the bases of the cylinder and specular reflection on the rest of the boundary. Furthermore, we work with polynomial, stretched exponential and inverse gaussian weights to construct the Cauchy theory near the equilibrium. We remark that all methods are quantitative thus all the constants are constructive and tractable.

The Boltzmann equation on smooth and cylindrical domains with Maxwell boundary conditions

TL;DR

This work studies the Boltzmann equation near its hydrodynamic limit on bounded domains with Maxwell boundary conditions, addressing both smooth domains and cylindrical geometries with a mix of diffusive and specular reflections. It develops a constructive, quantitative - theory by combining hypocoercivity with the stretching method, and handles both strongly and weakly confining weights, including spatially varying and discontinuous accommodation coefficients. The authors prove well-posedness and exponential relaxation to equilibrium in weighted spaces for small , using a rescaled formulation, a detailed regularity analysis in cylinders (Poisson and Lamé systems), and a careful operator splitting for weak confinement. The results extend prior cylindrical-domain analyses to variable accommodation, provide explicit decay rates with constructive constants, and offer a robust framework for the Boltzmann equation near the hydrodynamic limit in nontrivial geometries. This has potential implications for quantitative hydrodynamic limits and kinetic-fluid coupling in bounded domains with realistic boundary interactions.

Abstract

In this article we study the well-posedness of the Boltzmann equation near its hydrodynamic limit on a bounded domain. We consider two types of domains, namely domains with Maxwell boundary conditions where the accommodation coefficient is a continuous space dependent function for any , or cylindrical domains with diffusive reflection on the bases of the cylinder and specular reflection on the rest of the boundary. Furthermore, we work with polynomial, stretched exponential and inverse gaussian weights to construct the Cauchy theory near the equilibrium. We remark that all methods are quantitative thus all the constants are constructive and tractable.
Paper Structure (44 sections, 41 theorems, 343 equations)

This paper contains 44 sections, 41 theorems, 343 equations.

Table of Contents

  1. Introduction
  2. Framework
  3. Main result and discussion
  4. Transformation of the problem
  5. Strategy for the proof of the main result
  6. Organisation of the paper
  7. Notation
  8. Hypocoercivity
  9. Poisson equation in the cylinder
  10. Lamé system in the cylinder
  11. Preliminary inequalities
  12. Proof of Theorem \ref{['thm:LameRegularity']}
  13. Proof of Theorem \ref{['theo:Hypo']}
  14. Stretching method for strongly confining weights in smooth domains
  15. Auxiliary problem in finite time and backwards trajectories
  16. ...and 29 more sections

Key Result

Theorem 1.1

Consider either Assumption item:A1 or Assumption item:A2 to hold, and et $\omega$ be an admissible weight function. There exists ${\varepsilon}_0>0$ such that for every ${\varepsilon} \in (0, {\varepsilon}_0)$ there is $\eta({\varepsilon})\in (0,1)$, satisfying $\eta({\varepsilon})\to 0$ as ${\varep there exists a function $F\in L^\infty_\omega({\mathcal{U}})$ unique global solution to the Boltzm

Theorems & Definitions (79)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • ...and 69 more