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The Boltzmann equation for a multi-species inelastic mixture

Thomas Rey, Tommaso Tenna

Abstract

A granular gas is a collection of macroscopic particles that interact through energy-dissipating collisions, also known as inelastic collisions. This inelasticity is characterized by a collision mechanics in which mass and momentum are conserved and kinetic energy is dissipated. Such a system can be described by a kinetic equation of the Boltzmann type. Nevertheless, due to the macroscopic aspect of the particles, any realistic description of a granular gas should be written as a mixture model composed of M different species, each with its own mass. We propose in this work such a granular multi-species model and analyse it, providing Povzner-type inequalities, and a Cauchy theory in general Orlicz spaces. We also analyse its large time behavior, showing that it exhibits a mixture analogue of the seminal Haff's Law.

The Boltzmann equation for a multi-species inelastic mixture

Abstract

A granular gas is a collection of macroscopic particles that interact through energy-dissipating collisions, also known as inelastic collisions. This inelasticity is characterized by a collision mechanics in which mass and momentum are conserved and kinetic energy is dissipated. Such a system can be described by a kinetic equation of the Boltzmann type. Nevertheless, due to the macroscopic aspect of the particles, any realistic description of a granular gas should be written as a mixture model composed of M different species, each with its own mass. We propose in this work such a granular multi-species model and analyse it, providing Povzner-type inequalities, and a Cauchy theory in general Orlicz spaces. We also analyse its large time behavior, showing that it exhibits a mixture analogue of the seminal Haff's Law.

Paper Structure

This paper contains 26 sections, 22 theorems, 203 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Multi-species Granular Gases Equation
  3. Microscopic dynamics
  4. Geometry of inelastic collisions for mixtures
  5. Weak form
  6. Mono-species Collision Operator
  7. Bi-species Collision Operator
  8. Strong form
  9. Conservation properties
  10. Macroscopic properties of the granular gases mixtures operator
  11. The Homogeneous Equation
  12. Functional Framework
  13. Entropy for Multi-species Granular Gases
  14. Kinetic Energy Dissipation
  15. A Povzner-type Inequality for Granular Mixtures
  16. ...and 11 more sections

Key Result

Proposition 3.1

Let $(f_i^0)$ be a set of nonnegative distribution functions satisfying hypothesis_initial_moments and let $f_i(t,v)$ be the associated solution to the Cauchy problem homogeneous_BEs, with a constant restitution coefficient. Then in the VHS case,

Figures (1)

  • Figure 1: Geometrical configuration of the inelastic collision between two particles with different masses in the phase space.

Theorems & Definitions (37)

  • Remark 2.1
  • Proposition 3.1
  • Remark 3.2
  • Lemma 4.1: gamba2004, Lemma $3.1$
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • ...and 27 more