Global spherically symmetric solutions to the isothermal compressible Navier-Stokes equations with far-field vacuum

TL;DR

This paper proves the global existence and uniqueness of spherically symmetric strong solutions to 3D compressible Navier-Stokes equations with density-dependent viscosity and far-field vacuum, removing previous restrictions on parameters.

Contribution

It establishes the first global strong solutions for 3D spherically symmetric Navier-Stokes equations with density-dependent viscosity and vacuum, relaxing earlier parameter constraints.

Findings

Proves global existence and uniqueness of solutions.

Removes previous restrictions on parameters.

First such result for 3D spherically symmetric case.

Abstract

In this paper, we consider the global spherically symmetric strong solutions to the compressible Navier-Stokes equations with far-field vacuum and density-dependent degenerate viscosity, following the framework proposed by Bresch-Vasseur-Yu \cite{B-V-Y 2021}. For the 1D Navier-Stokes equations, Wen-Zhang \cite{W-Z SIAM 2025} considered the Cauchy problem which established the dependence relationship $\gamma-\delta-\frac{1}{p}\ge0$ within the $W^{2,p}(\mathbb{R})$ and $p\ge 2$. In this paper, we establish the global existence and uniqueness of strong solutions in $H^2([a,+\infty))$, $a>0$. In particular, we remove the restriction relating ($\gamma$, $\delta$, $p$), and instead assume that $\delta > 0.7427$. This result can be regarded as the first one on spherically symmetric strong solutions to the 3D Navier-Stokes equations with density-dependent viscosity proposed in \cite{B-V-Y 2021} and far-field vacuum.