Global Existence of Smooth Solutions and Convergence to Barenblatt Solutions for the Physical Vacuum Free Boundary Problem of Compressible Euler Equations with Damping

TL;DR

This paper proves the global existence and convergence of smooth solutions to the physical vacuum free boundary problem for 1D compressible Euler equations with damping, showing they tend to Barenblatt solutions with explicit rates.

Contribution

It establishes the global existence and detailed convergence rates of solutions near vacuum boundaries, introducing new analytical techniques for this class of problems.

Findings

Solutions exist globally in time for small perturbations of Barenblatt solutions.

Solutions converge to Barenblatt solutions with explicit rates.

Physical vacuum boundaries expand at a precise rate.

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 one-dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self-similar solution for the the porous media equation with the same total mass when the initial data is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in supereme norm and the precise expanding rate of the physical vacuum boundaries are also given. The proof is based on a construction of higher-order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time, an introduction of a new ansatz, higher-order nonlinear energy estimates and elliptic estimates.