Conservative Formulations of the Standard Enskog and Povzner Equations

TL;DR

This paper develops conservative formulations for the Standard Enskog and Povzner equations, expressing collision integrals as divergences of mass, momentum, and energy currents, extending previous Boltzmann equation results to dense gases.

Contribution

It introduces a conservative formulation of the Enskog and Povzner equations, representing collision integrals as phase-space divergences, extending Villani's Boltzmann equation results to dense gases.

Findings

Collision integrals expressed as divergence of mass, momentum, and energy currents.

Extension of Villani's Boltzmann equation results to dense gases.

Provides a framework for conservative formulations in kinetic theory.

Abstract

This article introduces a conservative formulation of the Standard Enskog equation and the Povzner equation, both of which generalize the Boltzmann equation by incorporating the contribution of particle volume in collisions. The primary result expresses these collision integrals as the divergence with respect to the velocity variable v of a mass current. Moreover, the terms v C[f,f] and |v|^2 C[f,f], where C[f,f] denotes the Standard Enskog or Povzner collision integral, are represented as phase-space divergences (that is, divergences in both position and velocity) of corresponding momentum and energy currents. This work extends Villani's earlier result (Math. Modelling Numer. Anal. M2AN 33 (1999), 209--227) for the classical Boltzmann equation to the case of dense gases.