The compressible viscous surface-internal wave problem: local well-posedness

TL;DR

This paper establishes local well-posedness for a free boundary problem involving two compressible, viscous fluid layers with moving interfaces, considering gravity and surface tension effects, using energy methods in Sobolev spaces.

Contribution

It proves local well-posedness for a complex free boundary problem involving compressible viscous fluids with moving interfaces, incorporating gravity and surface tension effects.

Findings

Proved local well-posedness of the free boundary problem.

Developed energy methods in Sobolev spaces for analysis.

Handled cases with and without surface tension.

Abstract

This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and the upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces. We prove that the problem is locally well-posed. Our method relies on energy methods in Sobolev spaces for a collection of related linear and nonlinear problems.