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

Abstract:We consider 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 Mach number, and they are uniform in surface tension coefficient under the Rayleigh-Taylor sign condition. 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.