Global Well-Posedness of Classical Solutions to the Multi-Dimensional Degenerate Compressible Navier-Stokes Equations with Large Spherically Symmetric Initial Data

TL;DR

This paper proves the global existence and uniqueness of classical solutions to multi-dimensional degenerate compressible Navier-Stokes equations with large spherically symmetric initial data, under specific conditions on viscosity and adiabatic index.

Contribution

It establishes the first global well-posedness results for classical solutions with large initial data in the degenerate viscosity setting, including new parameter ranges and vacuum prevention.

Findings

Global classical solutions exist for large spherically symmetric initial data.

Solutions do not develop vacuum if initially vacuum-free.

Extended parameter ranges for viscosity and adiabatic index are identified.

Abstract

This paper is concerned with the global existence and uniqueness of classical solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients in three-dimensional bounded domains or in the whole space $\mathbb{R}^N$ $(N=2,3)$ with non-vacuum far-field density. Specifically, we assume that the shear viscosity coefficient $\mu(\rho)=\rho^\alpha$ and the bulk viscosity coefficient $\lambda(\rho)=(\alpha-1)\rho^\alpha$, which satisfy the BD entropy relation. For arbitrarily large spherically symmetric initial data, we establish the global existence and uniqueness of spherically symmetric classical solutions under the following conditions: for $N=2$, $\alpha \in (0.54369,1)$ and $\gamma \in (1,\infty)$; for $N=3$ (both bounded domains and the whole space), $\alpha \in (0.67661,1)$ and $\gamma \in (1,6\alpha-3)$. In the two-dimensional case $\mathbb{R}^2$, the restriction on $\alpha$ can be further relaxed to $\alpha \in (9-6\sqrt{2},1)$ provided that the initial data satisfy additional weighted integrability conditions. Moreover, we show that the solution will not exhibit vacuum in any finite time provided that no vacuum is present initially.