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On the local Maxwellians solving the Boltzmann equation with boundary condition

Théophile Dolmaire

TL;DR

This work characterizes all local Maxwellians that solve the Boltzmann equation inside an open domain and under two common boundary conditions, bounce-back and specular reflection. By solving the free transport equation for a Maxwellian ansatz, the authors derive a complete affine-in-$x$ structure for the velocity field and obtain explicit time- and space-dependent coefficients, culminating in a closed-form expression for admissible local Maxwellians. They prove that BBBC permits only global Maxwellians (zero bulk velocity) irrespective of domain boundedness, while SRBC admits a rich, geometry-dependent classification in both $d=2$ and $d=3$; the allowable local Maxwellians are given explicitly for each boundary geometry, from simple planes to disks, cylinders, spheres, and helically symmetric surfaces. The results illuminate how boundary symmetries constrain transport equilibria and provide concrete formulas for use in long-time asymptotics and boundary-layer analyses.

Abstract

We derive the expressions of the local Maxwellians that solve the Boltzmann equation in the interior of an open domain. We determine which of these local Maxwellians satisfy the Boltzmann equation in a regular domain with boundary, without assuming the boundedness of the domain. We investigate separately, on the one hand, the case of the bounce-back boundary condition in any dimension, and on the other hand the case of the specular reflection boundary condition, in dimension $d = 2$ and $d = 3$. In the case of the bounce-back boundary condition, we prove that the only local Maxwellians solving the Boltzmann equation with boundary condition are the global Maxwellians. In the case of the specular reflection, we provide a complete classification of the domains for which only the global Maxwellians solve the Boltzmann equation with boundary condition, and we describe all the local Maxwellians that solve the equation for the domains presenting symmetries.

On the local Maxwellians solving the Boltzmann equation with boundary condition

TL;DR

This work characterizes all local Maxwellians that solve the Boltzmann equation inside an open domain and under two common boundary conditions, bounce-back and specular reflection. By solving the free transport equation for a Maxwellian ansatz, the authors derive a complete affine-in- structure for the velocity field and obtain explicit time- and space-dependent coefficients, culminating in a closed-form expression for admissible local Maxwellians. They prove that BBBC permits only global Maxwellians (zero bulk velocity) irrespective of domain boundedness, while SRBC admits a rich, geometry-dependent classification in both and ; the allowable local Maxwellians are given explicitly for each boundary geometry, from simple planes to disks, cylinders, spheres, and helically symmetric surfaces. The results illuminate how boundary symmetries constrain transport equilibria and provide concrete formulas for use in long-time asymptotics and boundary-layer analyses.

Abstract

We derive the expressions of the local Maxwellians that solve the Boltzmann equation in the interior of an open domain. We determine which of these local Maxwellians satisfy the Boltzmann equation in a regular domain with boundary, without assuming the boundedness of the domain. We investigate separately, on the one hand, the case of the bounce-back boundary condition in any dimension, and on the other hand the case of the specular reflection boundary condition, in dimension and . In the case of the bounce-back boundary condition, we prove that the only local Maxwellians solving the Boltzmann equation with boundary condition are the global Maxwellians. In the case of the specular reflection, we provide a complete classification of the domains for which only the global Maxwellians solve the Boltzmann equation with boundary condition, and we describe all the local Maxwellians that solve the equation for the domains presenting symmetries.
Paper Structure (16 sections, 17 theorems, 168 equations)

This paper contains 16 sections, 17 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Notations.
  3. Assumptions on the domain
  4. Obtaining the expressions of the admissible local Maxwellians
  5. Deriving a system on $\rho$, $a$ and $u$
  6. Obtaining a first expression for $u$
  7. Obtaining a second expression for $u$, and a first expression for $\rho$
  8. The general expressions of the coefficients $\sigma_0$, $C$ and $\rho_0$
  9. Determining $\sigma_0$.
  10. Determining $C$.
  11. Determining $\rho_0$.
  12. Concluding the proof concerning the expression of the local Maxwellians
  13. Local Maxwellians and boundary condition I: the bounce-back case
  14. Local Maxwellians and boundary condition II: the specular reflection case
  15. Local Maxwellians and specular reflection, in dimension $d = 2$
  16. ...and 1 more sections

Key Result

Theorem 1

Let $m : [0,T] \times \Omega \times \mathbb{R}^d \rightarrow \mathbb{R}_+$ be a local Maxwellian of the form: with $\rho$, $a$ and $u$ differentiable and $u$ twice differentiable in $x$, with $\rho$ and $a$ positive almost everywhere, that solves on $]0,T[\times\Omega\times\mathbb{R}^d$ the free transport equation: Then, there exist four real number $r_0$, $\alpha$, $\beta$ and $\gamma \in \math

Theorems & Definitions (46)

  • Theorem 1: Local Maxwellians solving the free transport equation
  • Remark 1
  • Lemma 1: System solved by the coefficients $\rho$, $a$ and $u$
  • Corollary 1: First simplification of the coefficient $a$
  • Lemma 2: First expression of the velocity field $u$
  • Remark 2
  • Lemma 3: Second expression of the velocity field $u$
  • Lemma 4: First expression of the density $\rho$
  • Lemma 5: Complete expression of the coefficient $a$
  • Lemma 6: Complete expression of the vector $C$
  • ...and 36 more