Global large strong solutions to the radially symmetric compressible Navier-Stokes equations in 2D solid balls

TL;DR

This paper proves the global existence of radially symmetric strong solutions to the 2D compressible Navier-Stokes equations with density-dependent viscosity in solid balls, extending previous results to larger viscosity parameters and establishing uniform density bounds.

Contribution

It establishes the first global existence of classical solutions for radially symmetric compressible Navier-Stokes equations in 2D solid balls with Dirichlet boundary conditions, for a wider range of viscosity parameters.

Findings

Global strong solutions exist for eta > 1.

Density remains uniformly bounded over time for certain eta.

Extends previous results to larger eta in 2D solid ball setting.

Abstract

In this paper, we consider the initial-boundary value problems of the compressible isentropic Navier-Stokes equations with density-dependent viscosity on two dimensional solid balls which was first introduced by Kazhikhov where shear viscosity $\mu$ is assumed to be constant and the bulk viscosity $\lambda$ is a polynomial of density up to power $\beta$. Under the condition of $\beta>1$, we prove the global existence of the radially symmetric strong solutions to the Kazhikhov models under Dirichlet boundary conditions for arbitrary large initial smooth data. Moreover, the density is shown to be uniformly bounded with respect to time when $\beta\in (\max\{1,\frac{\gamma+2}{4}\},\gamma]$. This improves the previous result of \cite{2016Huang,2022Huang} for general 2D domains where they require $\beta>4/3$ to ensure global existence and is the first result concerning the global existence of classical solutions to the radially symmetric compressible Navier-Stokes equations in 2D solid balls under Dirichlet boundary condition.