Large time behavior for the non‐isentropic Navier–Stokes–Maxwell system

Abstract

Communicated by C. MiaoIn this paper, we are concerned with the system of the non-isentropic compressible Navier-Stokes equations coupled with the Maxwell equations through the Lorentz force in three space dimensions. The global existence of solutions near constant steady states is established, and the time-decay rates of perturbed solutions are obtained. The proof for existence is due to the classical energy method, and the investigation of large-time behavior is based on the linearized analysis of the non-isentropic Navier-Stokes-Poisson equations and the electromagnetic part for the linearized isentropic NavierStokes-Maxwell equations. In the meantime, the time-decay rates obtained by Zhang, Li, and Zhu [J. Differential Equations, 250(2011), 866-891] for the linearized non-isentropic Navier-Stokes-Poisson equations are improved.

Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum

“…for any w ∈ H s (Ω) and s ≥ 1. It follows from (12), (18), (19), (20), (21), (22), (23) and (24) that…”

“…We also mention some works where the model seems a little bit different though it is also called compressible Navier-Stokes-Maxwell system. For the one-fluid nonisentropic compressible Navier-Stokes-Maxwell system in R 3 , the global existence of solutions near constant steady states is established and the time-decay rates of perturbed solutions are obtained in [23] with different right terms from those in this paper. The global existence of regular solutions to the 2D Navier-Stokes-Maxwell system is proved by using energy estimates and Brezis-Gallouet inequality, and a blow-up criterion for solutions to 3D Navier-Stokes-Maxwell system is obtained in [16].…”

“…Enlightened by the known results on well-posedness, we investigate the initial boundary value problem to the compressible non-isentropic Navier-Stokes-Maxwell system with vacuum in three space dimensions, where the right terms are different from those in [23]. The main difficulty in this paper is how to prove the estimate (8), which will be proved in section 2.…”

Global Existence of Solutions of the Navier-Stokes-Maxwell System in Besov Spaces

“…In 2016, Tan and Tong [19] studied the global existence and large time behavior of the compressible Navier-Stokes-Maxwell system in R 3 with linear damping. Liu and Su [14] obtained global existence near the constant steady states for the non-isentropic Navier-Stokes-Maxwell system. Ibrahim et al [18] established the existence and asymptotic stability for the time periodic small solutions of the incompressible Navier-Stokes-Maxwell system in the whole R 3 .…”

Global Well-posedness and Optimal Decay Rate of the Quasi-static Incompressible Navier–Stokes–Fourier–Maxwell–Poisson System

“…Moreover, Feng-Peng-Wang [7] proved the global well-posedness of the full CNSM system, Wang-Xu [33] gained the time decay rate of the global classical solutions. For more related works on the compressible hydrodynamic equations, we refer to [4,5,24,35] and the references cited therein.…”