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

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.