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This paper is concerned with the zero-relaxation limits for periodic smooth solutions of the non-isentropic Euler–Maxwell system in a three-dimensional torus prescribing the well/ill-prepared initial data. The non-isentropic Euler–Maxwell system can be reduced to a quasi-linear symmetric hyperbolic system of one order. By observing a special structure of the non-isentropic Euler–Maxwell system, we are able to decouple the system and develop a technique to achieve the a priori H s estimates, which guarantees the limit for the non-isentropic Euler–Maxwell system as the relaxation time τ → 0 . We realize that the convergence rate of the temperature is the same as the other unknowns in the L ∞ (0 , T 1 ; H s) , but the convergence rate of the temperature is slower than the velocity in L 2 (0 , T 1 ; H s) . The zero-relaxation limit presented here is the transport equation coupled with the drift–diffusion system. However, the limit of the isentropic Euler–Maxwell system is the classical drift–diffusion system. This shows the essential difference between the isentropic and non-isentropic Euler–Maxwell systems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the compressible Navier–Stokes–Maxwell equations with a non-constant background density in R 3 . We first show the existence and uniqueness of the non-trivial equilibrium (steady-state) of the system when the background density is a small variation of certain constant state, then we prove the asymptotic stability of the steady-state once the initial perturbation around the steady-state is small. Furthermore, by establishing the time-decay estimates for the corresponding linearized homogeneous equations, we artfully derive the time-algebraic convergence rates. The proof is based on the time-weighted energy method but with some new developments on the weight settings. [ABSTRACT FROM AUTHOR]

We study the stability of smooth solutions near non-constant equilibrium states for a bipolar full compressible Navier-Stokes-Maxwell system in a three-dimensional torus T=(R/Z)3. This system is quasilinear hyperbolic-parabolic. In the first part, by using the maximum principle, we find a non-constant steady state solution with small amplitude for this system. In the second part, with the help of suitable choices of symmetrizers and classic energy estimates, we prove that global smooth solutions exist and converge to the non-constant steady states as the time goes to infinity. As a byproduct, we obtain the global stability for the bipolar full compressible Navier-Stokes-Poisson system. [ABSTRACT FROM AUTHOR]