E. Carlen, R. Esposito, J. L. Lebowitz, R. Marra, A. Rokhlenko
Kinetics of a Model Weakly Ionized Plasma in the Presence of Multiple
Equilibria
(67K, TeX)
ABSTRACT. We study, globaly in time, the velocity distribution $f(v,t)$ of a
spatially homogeneous system that models a system of electrons in a
weakly ionized plasma, subjected to a constant external electric field
$E$. The density $f$ satisfies a Boltzmann type kinetic equation
containing a full nonlinear electron-electron collision term as well
as linear terms representing collisions with reservoir particles
having a specified Maxwellian distribution. We show that when the
constant in front of the nonlinear collision kernel, thought of as a
scaling parameter, is sufficiently strong, then the $L^1$ distance
between $f$ and a certain time dependent Maxwellian stays small
uniformly in $t$. Moreover, the mean and variance of this time
dependent Maxwellian satisfy a coupled set of nonlinear ODE's that
constitute the ``hydrodynamical'' equations for this kinetic system.
This remain true even when these ODE's have non-unique equilibria,
thus proving the existence of multiple stabe stationary solutions for
the full kinetic model. Our approach relies on scale independent
estimates for the kinetic equation, and entropy production
estimates. The novel aspects of this approach may be useful in other
problems concerning the relation between the kinetic and hydrodynamic
scales globably in time.