The Compressible Viscous Surface-Internal Wave Problem: Local Well-Posedness

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 in a functional framework suitable for our global analysis, which appears in a companion paper. Our method relies on energy methods in Sobolev spaces for a collection of related linear and nonlinear problems.