Global existence of the radially symmetric solutions of the Navier–Stokes equations for the isentropic compressible fluids

We study the isentropic compressible Navier–Stokes equations with radially symmetric data in an annular domain. We first prove the global existence and regularity results on the radially symmetric weak solutions with non‐negative bounded densities. Then we prove the global existence of radially symmetric strong solutions when the initial data ρ 0 , u 0 satisfy the compatibility condition$$-\mu \Delta {\rm \bf u}_{0} -(\lambda + \mu)\nabla \,{\rm div}\,{\rm \bf u}_{0} + \nabla (A\rho_{0}^{\gamma})=\rho_{0}^{1/2}{\rm \bf g}$$for some radially symmetric g ∈ L 2 . The initial density ρ 0 needs not be positive. We also prove some uniqueness results on the strong solutions. Copyright © 2004 John Wiley & Sons, Ltd.