A note on spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients

TL;DR

This paper establishes the existence of global weak solutions for spherically symmetric compressible Navier-Stokes equations with density-dependent viscosity, including applications to shallow water models, using approximate solutions.

Contribution

It provides the first proof of global weak solutions for these equations with density-dependent viscosity in the whole space or exterior domains.

Findings

Existence of global weak solutions for spherically symmetric flows.

Construction of approximate solutions to prove existence.

Application to the Saint-Venant shallow water model.

Abstract

In this note, by constructing suitable approximate solutions, we prove the existence of global weak solutions to the compressible Navier-Stokes equations with density-dependent viscosity coefficients in the whole space $\mathbb{R}^N$, $N\geq2$ (or exterior domain), when the initial data are spherically symmetric. In particular, we prove the existence of spherically symmetric solutions to the Saint-Venant model for shallow water in the whole space (or exterior domain).