On the Motion of a Compressible Gravity Water Wave with Vorticity

TL;DR

This paper establishes a priori energy estimates for the compressible Euler equations modeling water waves with vorticity and a free boundary, demonstrating convergence to incompressible solutions and advancing understanding of compressible water wave dynamics.

Contribution

It generalizes energy estimate methods to unbounded domains for compressible water waves with vorticity, including convergence results and initial data propagation.

Findings

First a priori energy bounds for compressible water waves

Uniform energy estimates in sound speed κ

Convergence of compressible to incompressible solutions

Abstract

We prove a priori estimates for the compressible Euler equations modeling the motion of a liquid with moving physical vacuum boundary in an unbounded initial domain. The liquid is under influence of gravity but without surface tension. Our fluid is not assumed to be irrotational. But the physical sign condition needs to be assumed on the free boundary. We generalize the method used in \cite{LL} to prove the energy estimates in an unbounded domain up to arbitrary order. To our knowledge, this result appears to be the first that concerns a priori energy bounds for the compressible water wave. In addition to that, the a priori energy estimates are in fact uniform in the sound speed $\kappa$. As a consequence, we obtain the convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of \cite{LL} to when you have an unbounded domain. On the other hand, we prove that there are initial data satisfying the compatibility condition in some weighted Sobolev spaces, and this will propagate within a short time interval, which is essential for proving long time existence for slightly compressible irrotational water waves.