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

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