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

TL;DR

This paper investigates the regularity of the solution map for the Euler-Poisson system in Sobolev spaces, revealing it is nowhere locally uniformly continuous, while ion trajectories are analytic curves.

Contribution

It demonstrates the lack of local uniform continuity of the solution map for the Euler-Poisson system and proves ion trajectories are analytic, using a geometric approach.

Findings

Solution map is nowhere locally uniformly continuous.

Ion trajectories are analytic curves.

Results hold for Sobolev spaces with s > 5/2.

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 $H^s(\R^3)$, $s > 5/2$. 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 $\R^3$.