DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

VLASOV-ENSKOG EQUATION WITH THERMAL
BACKGROUND IN GAS DYNAMICS
WILLIAM GREENBERG AND PENG LEI
In order to describe dense gases, a smooth attractive tail is added to the hard core repulsion
of the Enskog equation, along with a velocity diffusion. The existence of global-intime
renormalized solutions to the resulting diffusive Vlasov-Enskog equation is proved
for L 1 initial conditions.
Copyright © 2006 W. Greenberg and P. Lei. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
One of the problems of greatest current interest in the kinetic theory of classical systems
is the construction and analysis of systems which describe dense gases. The Boltzmann
equation, employed for more than a century to give the time evolution of gases, is accurate
only in the dilute-gas regime, yielding transport coefficients of an ideal fluid. In 1921,
Enskog introduced a Boltzmann-like collision process with hard core interaction, representing
molecules with nonzero diameter. The Enskog equation, as revised in the 1970’s
in order to obtain correct hydrodynamics, describes a nonideal fluid with transport coefficients
within 10% of those of realistic numerical models up to one-half close packing
density. A limitation, however, in its usefulness is that, unlike the Boltzmann equation,
no molecular interaction is modeled beyond the hard sphere collision.
A strategy to rectify this is the addition of a smooth attractive tail to a hard repulsive
core of radius a, thereby approximating a van der Waals interaction. The potential must
be introduced at the Liouville level. Following the pioneering work of de Sobrino [1],
Grmela [5, 6], Karkhech and Stell [7, 8], van Beijeren [11], and van Beijeren and Ernst
[12], with a potential satisfying the Poisson equation, one obtains the coupled kinetic
equations:
[ ∂
⃗r]
∂t +⃗v ·∇ f (⃗r,⃗v,t) =−⃗E ·∇ ⃗v f (⃗r,⃗v,t)+C E ( f , f ),
∫
div ⃗r
⃗E(⃗r,t) =− d⃗v f(⃗r,⃗v,t)
(1.1)
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 423–432