Spherically symmetric strong solution of compressible flow with large data and density-dependent viscosities

Abstract

We consider the isentropic compressible Navier-Stokes equations with density-dependent viscosities $\mu \left(\rho \right)={\rho }^{\alpha }$, $\lambda \left(\rho \right)=(\alpha -1){\rho }^{\alpha }$ in N-dimensional ($N=2,3$) bounded domain when the initial data are spherically symmetric. Based on the exploitation of the one-dimensional and non-swirl feature of symmetric solution, together with the BD-entropy estimates, the global well-posedness of strong solution with the symmetry center is proved for non-vacuum and large initial data as $N=2$, $\frac{4}{5}\le \alpha <1$, $1<\gamma$ or $N=3$, $\frac{7}{8}\le \alpha <1$, $1<\gamma <9\alpha -6$. In particular, it is shown that the solution will not develop the vacuum states in any finite time provided that no vacuum states are present initially.