Global well-posedness for 2D compressible radially symmetric Navier-Stokes equations with swirl

TL;DR

This paper establishes the global existence and asymptotic behavior of large strong solutions for 2D radially symmetric compressible Navier-Stokes equations with swirl, covering cases with vacuum and different bulk viscosity exponents.

Contribution

It provides the first global existence results for large strong solutions in 2D compressible Navier-Stokes with real boundary conditions for eta ≥ 1 and for eta in (0,1) under certain initial conditions.

Findings

Proved uniform boundedness of density independent of time.

Established global existence of large strong solutions for eta ≥ 1.

Extended results to cases with eta in (0,1) under initial density bounds.

Abstract

In this paper, we consider the radially symmetric compressible Navier-Stokes equations with swirl in two-dimensional disks, where the shear viscosity coefficient \(\mu = \text{const}> 0\), and the bulk one \(\lambda = \rho^\beta(\beta>0)\). When \(\beta \geq 1\), we prove the global existence and asymptotic behavior of the large strong solutions for initial values that allow for vacuum. One of the key ingredients is to show the uniform boundedness of the density independent of the time. When \(\beta\in(0,1)\), we prove the same conclusion holds when the initial value satisfies \(\norm{\rho_0}_{L^\infty} \leq a_0\), where \(a_0\) is given by \eqref{def a_0} as in Theorem \ref{Thm3}. To the best of our knowledge, this is the first result on the global existence of large strong solutions for 2D compressible Navier-Stokes equation with real non-slip (non Navier-slip) boundary conditions when $\beta\ge1$ and the first result on the global existence of strong solutions when $\beta\in(0,1)$