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

TL;DR

This paper proves the global existence and convergence of smooth solutions to a spherically symmetric gas-vacuum interface problem in compressible Euler equations with damping, showing they approach Barenblatt solutions over time.

Contribution

It establishes the global existence and detailed asymptotic behavior of solutions near vacuum boundaries for the first time in this context.

Findings

Solutions exist globally and remain smooth.

Solutions converge to Barenblatt self-similar solutions.

Explicit convergence rates and boundary expansion rates are provided.

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.