Global Solution to Enskog Equation with External Force in Infinite Vacuum

TL;DR

This paper establishes the existence and uniqueness of a global mild solution to the Enskog equation with an external force in an infinite vacuum, addressing complex collision dynamics in dense gases.

Contribution

It introduces new hypotheses for bicharacteristic equations of the Enskog equation and proves global solvability in an infinite vacuum setting for dense gases.

Findings

Existence and uniqueness of global mild solutions are proven.

New hypotheses for bicharacteristic equations are developed.

Analysis of collision integrals for dense gases is conducted.

Abstract

We first give hypotheses of the bicharacteristic equations corresponding to the Enskog equation with an external force. Since the collision operator of the Enskog equation is more complicated than that of the Boltzmann equation, these hypotheses are more complicated than those given by Duan et al. for the Boltzmann equation. The hypotheses are very related to collision of particles of moderately or highly dense gases along the bicharacteristic curves and they can be used to make the estimation of the so-called gain and loss integrals of the Enskog integral equation. Then, by controlling these integrals, we show the existence and uniqueness of the global mild solution to the Enskog equation in an infinite vacuum for moderately or highly dense gases. Finally, we make some remarks on the locally Lipschitz assumption of the collision factors in the Enskog equation.