Density-valued solutions for the Boltzmann-Enskog process

TL;DR

This paper proves the existence of density-valued solutions for the Boltzmann-Enskog process, a stochastic model describing the evolution of dense gases, using advanced functional analysis techniques.

Contribution

It establishes the existence and regularity of density solutions for the Boltzmann-Enskog process via Besov space analysis, extending previous stochastic process studies.

Findings

Existence of marginal probability density functions at each time.

Density functions reside in Besov spaces.

Velocity distribution support covers all of ^3.

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, R\"udiger and Sundar (2017) and further studied by Friesen, R\"udiger 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 $\mathbb{R}^3$.