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

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.