Time-Asymptotics of Physical Vacuum Free Boundaries for Compressible Inviscid Flows with Damping

TL;DR

This paper characterizes the long-term behavior of vacuum boundaries in compressible inviscid flows with damping, showing they asymptotically resemble Barenblatt solutions, thus providing a comprehensive understanding of their large-time dynamics.

Contribution

It establishes the first rigorous description of the large-time asymptotics of physical vacuum boundaries for compressible inviscid fluids, linking them to Barenblatt solutions.

Findings

Vacuum boundaries asymptotically match Barenblatt solutions.

Provides a complete description of large-time behavior.

First results on vacuum boundary asymptotics for inviscid fluids.

Abstract

In this paper, we prove the leading term of time-asymptotics of the moving vacuum boundary for compressible inviscid flows with damping to be that for Barenblatt self-similar solutions to the corresponding porous media equations obtained by simplifying momentum equations via Darcy's law plus the possible shift due to the movement of the center of mass, in the one-dimensional and three-dimensional spherically symmetric motions, respectively. This gives a complete description of the large time asymptotic behavior of solutions to the corresponding vacuum free boundary problems. The results obtained in this paper are the first ones concerning the large time asymptotics of physical vacuum boundaries for compressible inviscid fluids, to the best of our knowledge.