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

Abstract

We study the large-time asymptotic behavior of solutions toward the combination of a viscous contact wave with two rarefaction waves for the compressible non-isentropic Navier-Stokes equations coupling with the Maxwell equations through the Lorentz force (called the Navier-Stokes-Maxwell equations). It includes the electrodynamic effects into the dissipative structure of the hyperbolic-parabolic system and turns out to be more complicated than that in the simpler compressible Navier-Stokes equations. Based on a new observation of the specific structure of the Maxwell equations in the Lagrangian coordinates, we prove that this typical composite wave pattern is time-asymptotically stable for the Navier-Stokes-Maxwell equations under some smallness conditions on the initial perturbations and wave strength, and also under the assumption that the dielectric constant is bounded. The main result is proved by using elementary energy methods. This is the first result about the nonlinear stability of the combination of two different wave patterns for the compressible Navier-Stokes-Maxwell equations.

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

“…Huang-Liu in[13] 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 this in our paper, except for the similar dissipative term E + ub. Recently, Yao-Zhu in[49] study the asymptotic stability of the superposition of viscous contact wave with rarefaction waves for the compressible Navier-Stokes-Maxwell equations, which is the first result on the combination of two different wave patterns of this complex coupled model. But for the large-time behavior of solutions to an IBVP of the non-isentropic Navier-Stokes-Maxwell equations, as far as we know there are still few results.…”