Continuity of the flow map for symmetric hyperbolic systems and its application to the Euler--Poisson system

We show the continuity of the flow map for quasilinear symmetric hyperbolic systems with general right–hand sides in different functional setting, including weighted Sobolev spaces $H_{s,\delta }$. An essential tool to achieve the continuity of the flow map is a low regularity energy estimate. We then apply these results to the Euler–Poisson system which describes various systems of physical interests.