Kinetics of a model weakly ionized plasma in the presence of multiple equilibria

We study, globally 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 fully 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 L¹ 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 ordinary differential equations that constitute the "hydrodynamical" equations for this kinetic system. This remains true even when these ordinary differential equations have non-unique equilibria, thus proving the existence of multiple stable 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 globally in time.