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The global attractor of the inelastic linear Boltzmann equation in a gravity field for Maxwell molecules

Théophile Dolmaire, Nicola Miele, Alessia Nota

TL;DR

This work analyzes the linear inelastic Boltzmann equation for Maxwell molecules in a uniform gravity field, proving well-posedness in finite Radon measures and showing the existence of a unique non-equilibrium steady state for $0<r<1$ that globally attracts all measure-valued solutions with finite first moment. The authors deploy Markov-generator semigroup theory and Fourier-transform techniques (Bobylev method) to establish existence, uniqueness, and stability of steady states, along with detailed moment analyses for both the linear and rehomogenized collision operators. They further develop a comprehensive long-time framework based on Fourier-space comparison principles and construct explicit super-solutions to derive decay rates toward the steady state. The results illuminate the out-of-equilibrium behavior of inelastic Lorentz gases under gravity, providing a rigorous foundation for the global attractor structure and opening avenues for extending to more general collision kernels and nonlinear variants.

Abstract

In this article we consider the linear inelastic Boltzmann equation in presence of a uniform and fixed gravity field, in the case of Maxwell molecules. We first obtain a well-posedness result in the space of finite, non-negative Radon measures. In addition, we rigorously prove the existence of a stationary solution under the non-equilibrium condition which is induced by the presence of the external field. We further show that this stationary solution is unique in the class of the finite, non-negative Radon measures with finite first order moment, and that all the solutions in this class converge towards the stationary solution in the weak topology of the measures.

The global attractor of the inelastic linear Boltzmann equation in a gravity field for Maxwell molecules

TL;DR

This work analyzes the linear inelastic Boltzmann equation for Maxwell molecules in a uniform gravity field, proving well-posedness in finite Radon measures and showing the existence of a unique non-equilibrium steady state for that globally attracts all measure-valued solutions with finite first moment. The authors deploy Markov-generator semigroup theory and Fourier-transform techniques (Bobylev method) to establish existence, uniqueness, and stability of steady states, along with detailed moment analyses for both the linear and rehomogenized collision operators. They further develop a comprehensive long-time framework based on Fourier-space comparison principles and construct explicit super-solutions to derive decay rates toward the steady state. The results illuminate the out-of-equilibrium behavior of inelastic Lorentz gases under gravity, providing a rigorous foundation for the global attractor structure and opening avenues for extending to more general collision kernels and nonlinear variants.

Abstract

In this article we consider the linear inelastic Boltzmann equation in presence of a uniform and fixed gravity field, in the case of Maxwell molecules. We first obtain a well-posedness result in the space of finite, non-negative Radon measures. In addition, we rigorously prove the existence of a stationary solution under the non-equilibrium condition which is induced by the presence of the external field. We further show that this stationary solution is unique in the class of the finite, non-negative Radon measures with finite first order moment, and that all the solutions in this class converge towards the stationary solution in the weak topology of the measures.

Paper Structure

This paper contains 29 sections, 25 theorems, 176 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The model
  3. Lorentz gas and granular material: a brief review of the literature
  4. Existing results concerning conservative Lorentz gases.
  5. Qualitative behaviour of granular gases.
  6. Mathematical theory of the inelastic Boltzmann equation
  7. The non-linear inelastic Boltzmann equation.
  8. The linear inelastic Boltzmann equation.
  9. Notations
  10. Main results
  11. Well-posedness results
  12. Markov Generator and Markov Semigroup
  13. Proof of Theorem \ref{['thm:wp1']}
  14. Existence, Uniqueness and Stability of steady states
  15. Moments estimates
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $d = 2$ or $3$, and let $a \in \mathbb{R}^d$ be a fixed vector. Assume that the restitution coefficient $r$ satisfies $0 < r \leq 1$. Let $f_0\in \mathscr{M}_+(\mathbb{R}^d)$ and let $b(\vert N \cdot \omega\vert )$ satisfy Assumption ass:kernelB. Then, there exists a unique weak solution $f \in

Figures (1)

  • Figure 1: The phase portrait of the ODE system \ref{['EQUATSyste2dMmt_Cas_MaxwlRehomGammaGener']}. The vertical isocline is represented in blue, the horizontal isocline is represented in red. At the intersection between the two isoclines lies the only equilibrium of the system \ref{['EQUATSyste2dMmt_Cas_MaxwlRehomGammaGener']}.

Theorems & Definitions (56)

  • Definition 1: Weak solution of \ref{['eq:LinInBolstrong_Cauchy']}
  • Theorem 1.1: Well-posedness
  • Remark 1
  • Definition 2: Stationary solution to \ref{['eq:LinInBolstrong_Cauchy']}
  • Theorem 1.2: Existence and uniqueness of steady states
  • Theorem 1.3: Stability of steady states
  • Remark 2
  • Proposition 1.4: Long time-behaviour of the moments
  • Definition 3
  • Remark 3
  • ...and 46 more