Critical thresholds in pressureless Euler–Poisson equations with background states

Abstract

We investigate the critical threshold phenomena in a large class of one-dimensional pressureless Euler–Poisson (EP) equations, with non-vanishing background states. First, we establish local-in-time well-posedness in proper regularity spaces specifically involving ${\dot{W}}^{-1,p}$, which are adapted for a neutrality condition to hold. This negative homogeneous Sobolev regularity is shown to be necessary: we prove an ill-posedness result in classical Sobolev spaces $H^s(\mathbb{R})$ in the absence of this negative Sobolev regularity. Next we study the critical threshold phenomena in the neutrality-condition-satisfying pressureless EP systems. We prove that in the case of attractive forcing, the neutrality condition can further restrict the sub-critical region into its borderline, namely, the sub-critical region is reduced to a single line in the phase plane. We then turn to providing a rather definitive answer for the critical thresholds in the case of repulsive EP systems with variable backgrounds. As an application, we analyze the critical thresholds for the damped EP system for cold-plasma-ion dynamics, where the density of electrons is given by the Maxwell–Boltzmann relation.