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 made of ions with a negligible ion temperature) 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, $({\rho }_0,u_0)\mapsto (\rho (T),u(T))$, is nowhere locally uniformly continuous. On the other hand it turns out that the trajectories of the ions are analytic curves in ${\mathbb{R}}^3$.