On the regularity of the solution map of the Euler–Poisson system

Abstract

In this paper we consider the Euler–Poisson system (describing a plasma consisting of positive ions with anegligible temperature and massless electrons in thermodynamical equilibrium) on the Sobolev spaces $H^s(mathbb{R}^3),;s>5/2$. Using a geometric approach we show that for any time T > 0 the corresponding solution map, $(p_{0,u_0});mapsto(p(T),;u(T))$ is nowhere locally uniformly continuous. On the other hand it turns out that the trajectories of the ions are analyticcurves in $mathbb{R}^3$