KINETICS OF A MODEL WEAKLY IONIZED PLASMA IN THE PRESENCE OF MULTIPLEEQUILIBRIA

We study, globally in time, the velocity distribution f(upsilon, t) of a spatially homogeneous system that models a system of electrons in a weakly ionized plasma, subjected to a constant external electric fiel d E. The density f satisfies a Boltzmann-type kinetic equation contain ing a fully nonlinear electron-electron collision term as well as line ar terms representing collisions with reservoir particles having a spe cified Maxwellian distribution. We show that when the constant in fron t of the nonlinear collision kernel, thought of as a scaling parameter , is sufficiently strong, then the L-1 distance between f and a certai n time-dependent Maxwellian stays small uniformly in t. Moreover, the mean and variance of this time-dependent Maswellian satisfy a coupled set of nonlinear ordinary differential equations that constitute the ' 'hydrodynamical'' equations for this kinetic system. This remains true even when these ordinary differential equations have non-unique equil ibria, thus proving the existence of multiple stable stationary soluti ons for the full kinetic model. Our approach relies on scale-independe nt estimates for the kinetic equation, and entropy production estimate s. The novel aspects of this approach may be useful in other problems concerning the relation between the kinetic and hydrodynamic scales gl obally in time.