On uniqueness and stability for the Enskog equation

TL;DR

This paper establishes conditions for the uniqueness and stability of solutions to the Enskog equation, a fundamental model for dense gases, using Wasserstein distance inequalities and extending previous stochastic process analyses.

Contribution

It introduces a shifted distance method to prove Wasserstein inequalities, ensuring uniqueness and stability of solutions for the Enskog equation with general potentials.

Findings

Proved a Wasserstein distance inequality for measure solutions.

Established sufficient conditions for solution uniqueness.

Extended analysis to soft and hard potentials without angular cut-off.

Abstract

The time-evolution of a moderately dense gas in a vacuum is described in classical mechanics by a particle density function obtained from the Enskog equation. Based on a McKean-Vlasov stochastic equation with jumps, the associated stochastic process was recently studied in \cite{ARS17}. The latter work was extended in \cite{FRS18} to the case of general hard and soft potentials without Grad's angular cut-off assumption. By the introduction of a shifted distance that exactly compensates for the free transport term that accrues in the spatially inhomogeneous setting, we prove in this work an inequality on the Wasserstein distance for any two measure-valued solutions to the Enskog equation. As a particular consequence, we find sufficient conditions for the uniqueness and continuous-dependence on initial data for solutions to the Enskog equation applicable to hard and soft potentials without angular cut-off.