Stability of rarefaction wave for the compressible non-isentropic Navier-Stokes-Maxwell equations

Abstract

We study the large-time asymptotic behavior of solutions toward the rarefaction wave of the compressible non-isentropic Navier-Stokes equations coupling with Maxwell equations under some small perturbations of initial data and also under the assumption that the dielectric constant is bounded. For that, the dissipative structure of this hyperbolic-parabolic system is studied to include the effect of the electromagnetic field into the viscous fluid and turns out to be more complicated than that in the simpler compressible Navier-Stokes system. The proof of the main result is based on the elementary L 2 energy methods.

Asymptotic stability of the superposition of viscous contact wave with rarefaction waves for the compressible Navier-Stokes-Maxwell equations

“…In this paper, we shall restrict ourselves to the one-dimensional motion (see [3], [40]) on the whole spatial domain R:…”

“…For example, to the authors' best knowledge, there are only three relevant results. Luo-Yao-Zhu in [27] and Yao-Zhu in [40] established the stability of rarefaction wave for the isentropic Navier-Stokes-Maxwell equations and non-isentropic ones under small H 1 -initial perturbations, respectively. What' more, Huang-Liu in [14] consider the stability of rarefaction wave for a macroscopic model derived from the Vlasov-Maxwell-Boltzmann system, in which the model they consider is obviously different from ours in this paper, except for the similar dissipative term E + ub.…”

“…Finally, for the stability of the superposition of viscous contact wave with rarefaction waves to the compressible Navier-Stokes-Maxwell equations, we need to consider the wave interactions of these two types of basic waves, under the combined effects of the electric field and the magnetic field. Borrowing some similar computations from the stability of these two typical wave patterns to the compressible Navier-Stokes equations in [7], and combining with the arguments in [40] of the stability of a single rarefaction wave for the Navier-Stokes-Maxwell equations in the Eulerian coordinates, we can modify our method of a single viscous contact wave slightly to overcome the difficulties caused by the two rarefaction waves.…”

Large-time behavior of solutions to the outflow problem for the compressible Navier-Stokes-Maxwell equations

“…Since the dynamic motion of the fluid and the electromagnetic fields couple strongly, the governing system in the non-isentropic case is derived from fluid mechanics with appropriate modifications to take account of the electromagnetic effects, which consists of the laws of conservation of mass, momentum and energy, Maxwell's law, and the law of conservation of electric charge (see [14], [17]). In this paper, we shall restrict ourselves to the one-dimensional motion (see [4], [50]) on the half line R + :…”

“…To the authors' best knowledge, there are only four relevant results. To the Cauchy problem, Luo-Yao-Zhu [32] and Yao-Zhu [50]…”