Compressible Gravity-Capillary Water Waves with Vorticity: Local Well-Posedness, Incompressible and Zero-Surface-Tension Limits

TL;DR

This paper establishes local well-posedness for 3D compressible water waves with gravity and surface tension, allowing for vorticity, and analyzes incompressible and zero-surface-tension limits without regularity loss.

Contribution

It introduces a hyperbolic approach avoiding Nash-Moser iteration and achieves uniform energy estimates in Mach number and surface tension, enabling simultaneous limits.

Findings

Proved local well-posedness for compressible water waves with vorticity.

Established uniform energy estimates in Mach number and surface tension.

Derived incompressible and zero-surface-tension limits without regularity loss.

Abstract

We consider the 3D compressible isentropic Euler equations describing the motion of a liquid in an unbounded initial domain with a moving boundary and a fixed flat bottom at finite depth. The liquid is under the influence of gravity and surface tension, and it is not assumed to be irrotational. We prove local well-posedness by combining a carefully designed approximate system and a hyperbolic approach, which allows us to avoid using Nash-Moser iteration. The energy estimates yield no regularity loss and are uniform in both Mach number and surface tension coefficient, provided the Rayleigh-Taylor sign condition is satisfied. We thus simultaneously obtain incompressible and zero surface tension limits. Moreover, we can drop the uniform boundedness (with respect to Mach number) on high-order time derivatives by applying the paradifferential calculus to the analysis of the free-surface evolution.