We consider smooth solutions of the Euler‐Poisson system for ion dynamics in which the electron density is replaced by a Boltzmann relation. The system arises in the modeling of plasmas, where appear two small parameters, the relaxation time and the Debye length. When the initial data are sufficiently close to constant equilibrium states, we prove the convergence of the system for all time, as each of the parameters goes to zero. The limit systems are drift‐diffusion equations and compressible Euler equations. The proof is based on uniform energy estimates and compactness arguments.

In this paper, we consider a stability problem of a kind of boundary value problem for quasilinear hyperbolic systems. For small boundary data, we prove that the C 1 solution exists globally in time when the system is weakly linearly degenerate. In the special case of linear degeneracy, the smallness assumption on the boundary data is weakened. The error estimate in L ∞ T L 1 space between two different solutions with different boundary data is also obtained. In our proof, an important estimate which describes the interaction of different waves is established by constructing a continuous Glimm function. This estimate together with uniform a prior estimates with respect to the time leads to the global-in-time existence of smooth solutions. Finally, we apply the stability results to the isentropic Euler equations and the system of the motion of an elastic string.

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.

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