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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 consider stability problems for the compressible viscous and diffusive magnetohydrodynamic (MHD) equations arising in the modeling of magnetic field confinement nuclear fusion. In the first part, we investigate the Cauchy problem to the barotropic MHD system. With the help of the techniques of anti-symmetric matrix and an induction argument on the order of the space derivatives of solutions in energy estimates, we prove that smooth solutions exist globally in time near the non-constant equilibrium solutions. We also obtain the asymptotic behavior of solutions when the time goes to infinity. The result shows that gradients of both the velocity and the magnetic field converge to the equilibrium solutions with the same norm ‖ · ‖ H s - 3 , while the density converge with stronger norm ‖ · ‖ H s - 1 . In the second part, the initial value problem to the full MHD system is studied. By means of the techniques of choosing a non-diagonal symmetrizer and elaborate energy estimates, we prove the existence and uniqueness of global solutions to the system when the initial data are close to the non-constant equilibrium states. We find that both the density and temperature converge to the equilibrium states with the same norm ‖ · ‖ H s - 1 . These phenomena on the charge transport show the essential relationship of the equations between the barotropic and the full MHD systems. [ABSTRACT FROM AUTHOR]

In this article we consider a Cauchy problem for the full compressible Navier–Stokes–Maxwell system arising from viscosity plasmas. This system is quasilinear hyperbolic–parabolic. With the help of techniques of symmetrizers and the smallness of non-constant equilibrium solutions, we establish that global smooth solutions exist and converge to the equilibrium solution as the time approaches infinity. This result is obtained for initial data close to the steady-states. As a byproduct, we obtain the global stability of solutions near the equilibrium states for the full compressible Navier–Stokes–Poisson system in a three-dimensional torus. [ABSTRACT FROM AUTHOR]

This paper is concerned with the periodic problem to the two-fluid non-isentropic Euler–Maxwell (N-E-M) equations. The equations arises in the modeling of magnetic plasma, in which appear two physical parameters, the mass of an electron m e and the mass of an ion m i . With the help of methods of asymptotic expansions, we prove the local-in-time convergence of smooth solutions to this problem by setting m e = 1 and letting m i → + ∞ . Moreover, when the initial data are near constant equilibrium states, by means of uniform energy estimates and compactness arguments, we rigorously prove the infinity-ion-mass convergence of the system for all time. The limit system is the one-fluid N-E-M system. [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]