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Density-valued solutions for the Boltzmann-Enskog process

Christian Ennis, Barbara Rüdiger, Padmanabhan Sundar

TL;DR

This work proves that the velocity marginal of the Boltzmann-Enskog process, which models moderately dense gases via a McKean–Vlasov jump dynamics driven by a Poisson measure, possesses a genuine density in Besov spaces for any positive time. The authors construct an ε-regularized approximating process and establish convergence, moment and regularity estimates, and stochastic continuity; they also derive non-degeneracy and lower-bound properties of the velocity distribution through geometric and Fourier-analytic arguments. By leveraging the Debussche–Romito framework, they deduce the existence of a density in a Besov space $B^{a-\alpha}_{1,\infty}(\mathbb{R}^3)$ for suitable $a>\alpha$ under non-Dirac initial data with finite second moments. A key outcome is that the velocity support becomes all of $\mathbb{R}^3$ for $t>0$, providing strong regularity for the kinetic description of moderately dense gases. These results bridge stochastic representations with functional-analytic regularity, informing both the theory of measure-valued solutions and potential numerical approaches to dense-gas kinetics.

Abstract

The time evolution of moderately dense gas evolving in vacuum described by the Boltzmann-Enskog equation is studied. The associated stochastic process, the Boltzmann-Enskog process, was constructed by Albeverio, Rüdiger and Sundar (2017) and further studied by Friesen, Rüdiger and Sundar (2019, 2022). The process is given by the solution of a McKean-Vlasov equation driven by a Poisson random measure, the compensator depending on the distribution of the solution. The existence of a marginal probability density function at each time for the measure-valued solution is established here by using a functional-analytic criterion in Besov spaces Debussche and Romito (2014), and Fournier (2015). In addition to existence, the density is shown to reside in a Besov space. The support of the velocity marginal distribution is shown to be the whole of $\mathbb{R}^3$.

Density-valued solutions for the Boltzmann-Enskog process

TL;DR

This work proves that the velocity marginal of the Boltzmann-Enskog process, which models moderately dense gases via a McKean–Vlasov jump dynamics driven by a Poisson measure, possesses a genuine density in Besov spaces for any positive time. The authors construct an ε-regularized approximating process and establish convergence, moment and regularity estimates, and stochastic continuity; they also derive non-degeneracy and lower-bound properties of the velocity distribution through geometric and Fourier-analytic arguments. By leveraging the Debussche–Romito framework, they deduce the existence of a density in a Besov space for suitable under non-Dirac initial data with finite second moments. A key outcome is that the velocity support becomes all of for , providing strong regularity for the kinetic description of moderately dense gases. These results bridge stochastic representations with functional-analytic regularity, informing both the theory of measure-valued solutions and potential numerical approaches to dense-gas kinetics.

Abstract

The time evolution of moderately dense gas evolving in vacuum described by the Boltzmann-Enskog equation is studied. The associated stochastic process, the Boltzmann-Enskog process, was constructed by Albeverio, Rüdiger and Sundar (2017) and further studied by Friesen, Rüdiger and Sundar (2019, 2022). The process is given by the solution of a McKean-Vlasov equation driven by a Poisson random measure, the compensator depending on the distribution of the solution. The existence of a marginal probability density function at each time for the measure-valued solution is established here by using a functional-analytic criterion in Besov spaces Debussche and Romito (2014), and Fournier (2015). In addition to existence, the density is shown to reside in a Besov space. The support of the velocity marginal distribution is shown to be the whole of .

Paper Structure

This paper contains 6 sections, 20 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Approximating Process
  4. Establishing Support on $\mathbb{R}^3$
  5. Estimates for the Boltzmann-Enskog process
  6. Existence of Density for the Boltzmann-Enskog process

Key Result

Proposition 1.1

Let $\Psi(r,v) \in C_c(\mathbb{R}^6)$, as a function of $r,v \in \mathbb{R}^3$. With $B$ as in eqn1.8, we have

Theorems & Definitions (33)

  • Proposition 1.1
  • Lemma 2.1
  • Lemma 2.2: Lemma 3.1 from tanaka
  • Definition 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • proof
  • Lemma 3.1
  • Lemma 4.1: Lemma 4.1 from Fournier four
  • ...and 23 more