Global existence of smooth solutions to a full Euler–Poisson system in one space dimension

Abstract

When a strictly dissipative term arises in the energy equation, the Cauchy problem for Euler–Poisson systems admits global smooth solutions for small initial data. This was proved in previous studies. In this paper, we study a stability problem for a full Euler–Poisson system without dissipation in the energy equation. We prove the global existence of smooth solutions near constant equilibrium states in one space dimension. The stability results are obtained for both one-fluid and two-fluid Euler–Poisson systems. In our proof of these results, we show an L2 energy equality in Euler coordinates. Then, we establish energy estimates of derivatives of the solution in Lagrangian coordinates by a characteristic technique. These estimates together with the equivalence of smooth solutions between the two coordinates yield the global existence of the solution.