Stability of steady states of the compressible Euler-Poisson system in R3

We investigate the global existence and asymptotic behavior of smooth solutions near a non-flat steady state to the compressible Euler-Poisson system in R3. Using some concise energy estimates and an interpolation trick, we refine the global existence and show that the solution converges to the stationary solution exponentially fast. We assume that the H3 norms of the initial density and velocity are small, but the higher derivatives can be arbitrarily large. In this sense, our method simplifies the proof of Hsiao et al. (2003) [17] and improves the results of it.