Approximation of a compressible Euler-Poisson equations by a non-isentropic Euler-Maxwell equations

In this article, the convergence of time-dependent and non-isentropic Euler-Maxwell equations to compressible Euler-Poisson equations in a torus via the non-relativistic limit is studied. The local existence of smooth solutions to both equations is proved by using energy method for first order symmetrizable hyperbolic systems. The method of asymptotic expansion and the symmetric hyperbolic property of the systems are used to justify the convergence of the limit.