Global Solutions to the Gas-Vacuum Interface Problem of Isentropic Compressible Inviscid Flows with Damping in Spherically Symmetric Motions and Physical Vacuum

Abstract

For the physical vacuum free boundary problem with the sound speed being $C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous near vacuum boundaries of the three-dimensional compressible Euler equations with damping, the global existence of spherically symmetric smooth solutions is proved, which are shown to converge to Barenblatt self-similar solutions of the porous media equation with the same total masses when initial data are small perturbations of Barenblatt solutions. The pointwise convergence with a rate of density, the convergence rate of velocity in supreme norm and the precise expanding rate of physical vacuum boundaries are also given by constructing nonlinear functionals with space-time weights featuring the behavior of solutions in large time and near the vacuum boundary and the center of symmetry, the nonlinear energy estimates and elliptic estimates.