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

TL;DR

This paper proves the asymptotic stability of a combined wave pattern involving viscous contact and rarefaction waves for the complex compressible Navier-Stokes-Maxwell system, incorporating electrodynamic effects.

Contribution

It is the first to establish nonlinear stability of combined wave patterns for the Navier-Stokes-Maxwell equations using elementary energy methods.

Findings

Proved time-asymptotic stability under small initial perturbations.

Incorporated electrodynamic effects into the stability analysis.

Established stability for a composite wave pattern in a complex hyperbolic-parabolic system.

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.