Local Well-posedness for the Motion of a Compressible Gravity Water Wave with Vorticity

TL;DR

This paper establishes the local well-posedness of 3D compressible Euler equations for gravity-driven water waves with vorticity, using tangential smoothing and Alinhac's good unknowns, without requiring irrotationality or high regularity.

Contribution

It introduces a novel approach combining tangential smoothing and Alinhac's good unknowns to prove well-posedness without assuming irrotationality or high regularity.

Findings

Proves local well-posedness for 3D compressible water waves with vorticity.

Develops energy estimates uniform in smoothing parameter.

Avoids reliance on higher order wave equations and delicate elliptic estimates.

Abstract

In this paper we prove the local well-posedness (LWP) for the 3D compressible Euler equations describing the motion of a liquid in an unbounded initial domain with moving boundary. The liquid is under the influence of gravity but without surface tension, and it is not assumed to be irrotational. We apply the tangential smoothing method introduced in [10,11] to construct the approximation system with energy estimates uniform in the smooth parameter. It should be emphasized that, when doing the nonlinear a priori estimates, we need neither the higher order wave equation of the pressure and delicate elliptic estimates, nor the higher regularity on the flow-map or initial vorticity. Instead, we adapt the Alinhac's good unknowns to the estimates of full spatial derivatives.