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

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 $\beta\in(0,1)$