Global Spherically Symmetric Solutions and Relaxation Limit for the Relaxed Compressible Navier-Stokes Equations

TL;DR

This paper proves the global existence and relaxation limit of spherically symmetric solutions to a modified compressible Navier-Stokes system with Maxwell law in an exterior domain, using energy estimates and compactness methods.

Contribution

It constructs an approximate system with a non-characteristic boundary, establishes uniform a priori estimates, and rigorously justifies the relaxation limit for the system.

Findings

Global well-posedness of the system is established.

Uniform a priori estimates are derived for solutions.

The relaxation limit is rigorously justified.

Abstract

This paper studies an initial boundary value problem for the multidimensional hyperbolized compressible Navier-Stokes equations, in which the classical Newtonian law is replaced by the Maxwell law. We seek spherically symmetric solutions to the studied system in an exterior domain of a ball in $\mathbb R^3$, which are a system possessing a uniform characteristic boundary. First, we construct an approximate system featuring a non-characteristic boundary and establish its local well-posedness. Subsequently, by defining a suitable weighted energy functional and carefully handling boundary terms, we derive uniform a priori estimates, enabling the proof of uniform global existence. Leveraging these uniform estimates alongside standard compactness arguments, we establish the global well-posedness of the original system. Additionally, we rigorously justify the global relaxation limit.