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An Alexander-like theorem for a particle model with inelastic collisions

Théophile Dolmaire, Juan J. L. Velázquez

TL;DR

The paper analyzes a finite system of inelastic hard spheres with emission in two dimensions, where each energetic collision dissipates a fixed energy $\varepsilon_0$. It develops the Transport-Collision-Transport (TCT) framework to track how the Lebesgue measure in phase space evolves under free transport and binary collisions, and proves a key volume formula for the TCT flow. A central finding is that, in $d=2$, the scattering map preserves velocity-space Lebesgue measure, while the full flow need not be measure-preserving due to collision-time contributions, enabling an Alexander-type global well-posedness result for almost every initial datum. The authors adapt Alexander’s scheme to IHSE by introducing pathological sets and using a BuFK bound on elastic collisions to control the number of events, with a complementary low-energy direct proof for the single-inelastic-collision regime. The work highlights how measure-theoretic properties of the scattering interact with energy dissipation to yield global well-posedness in a dissipative, high-collision setting, and discusses limitations in higher dimensions or with fixed restitution models.

Abstract

We consider a finite system of hard spheres that collide inelastically according to a particular model, losing a fixed amount of kinetic energy at each collision. We develop the theory of the Transport-Collision-Transport (TCT) dynamics, which allows to study precisely the evolution of the Lebesgue measure in the phase space under the action of the flows of particle systems that can interact via instantaneous binary collision. We show that the scattering mapping associated to the inelastic hard sphere system we introduce preserves locally the Lebesgue measure in the velocity space, in spite of the fact that a positive amount of kinetic energy is lost at each inelastic collision. We prove the analog of Alexander's theorem for our model, which allows us to deduce the global well-posedness of the trajectories, for almost every initial datum.

An Alexander-like theorem for a particle model with inelastic collisions

TL;DR

The paper analyzes a finite system of inelastic hard spheres with emission in two dimensions, where each energetic collision dissipates a fixed energy . It develops the Transport-Collision-Transport (TCT) framework to track how the Lebesgue measure in phase space evolves under free transport and binary collisions, and proves a key volume formula for the TCT flow. A central finding is that, in , the scattering map preserves velocity-space Lebesgue measure, while the full flow need not be measure-preserving due to collision-time contributions, enabling an Alexander-type global well-posedness result for almost every initial datum. The authors adapt Alexander’s scheme to IHSE by introducing pathological sets and using a BuFK bound on elastic collisions to control the number of events, with a complementary low-energy direct proof for the single-inelastic-collision regime. The work highlights how measure-theoretic properties of the scattering interact with energy dissipation to yield global well-posedness in a dissipative, high-collision setting, and discusses limitations in higher dimensions or with fixed restitution models.

Abstract

We consider a finite system of hard spheres that collide inelastically according to a particular model, losing a fixed amount of kinetic energy at each collision. We develop the theory of the Transport-Collision-Transport (TCT) dynamics, which allows to study precisely the evolution of the Lebesgue measure in the phase space under the action of the flows of particle systems that can interact via instantaneous binary collision. We show that the scattering mapping associated to the inelastic hard sphere system we introduce preserves locally the Lebesgue measure in the velocity space, in spite of the fact that a positive amount of kinetic energy is lost at each inelastic collision. We prove the analog of Alexander's theorem for our model, which allows us to deduce the global well-posedness of the trajectories, for almost every initial datum.
Paper Structure (14 sections, 12 theorems, 158 equations)

This paper contains 14 sections, 12 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. The inelastic hard spheres with emission
  3. The model
  4. Definition of the Transport-Collision-Transport (TCT) dynamics
  5. Definition of the dynamics, without collision.
  6. Definition of the dynamics, with a single collision (TCT dynamics).
  7. Evolution of the measure of the phase space under the action of the TCT dynamics
  8. Applications of the TCT volume formula \ref{['EQUATTheo1EvoluMesur_TCT_']}
  9. The TCT volume formula, in the case of elastic hard spheres
  10. The TCT volume formula, in the case of inelastic hard spheres with emission
  11. Consequences and interpretation of Theorem \ref{['THEORSect2ConservatiMesur']}
  12. Global well-posedness of the flow of the inelastic hard spheres with emission
  13. Direct proof of Theorem \ref{['THEORSect3AlexanderIHSWEM']} in the case of a single inelastic collision
  14. Interpretation of the result of Theorem \ref{['THEORSect2ConservatiMesur']}

Key Result

Theorem 1

Let $\tau > 0$ be a positive real number. Let us consider a general TCT dynamics $\mathcal{T}_\tau$, defined as in Definition DEFIN_TCT_ProceInelaHSwEm on the time interval $]0,\tau]$ as the composition of the four functions functions $\varphi_1^{(i,j)}$, $\varphi_2^{(i,j)}$, $\widetilde{\varphi}_3^ where $N$ is the Jacobian matrix of the post-collisional velocity $V'_N = (v'_1,\dots,v'_N)$ with r

Theorems & Definitions (45)

  • Definition 1: Phase space
  • Remark 1
  • Definition 2: Collision time function, collision times
  • Definition 3: Grazing collision discriminant
  • Definition 4: Domain of the free dynamics, definition of the IHSE dynamics $\mathcal{T}_\tau$ (I)
  • Definition 5: Domain of the TCT dynamics
  • Remark 2
  • Definition 6: Elementary functions of the TCT dynamics
  • Remark 3
  • Definition 7: TCT dynamics, definition of the IHSE dynamics $\mathcal{T}_\tau$ (II)
  • ...and 35 more