Interface evolution: water waves in 2-D

TL;DR

This paper proves local existence for 2-D water wave equations with free boundary evolution between two fluids, under the Rayleigh-Taylor condition, using Sobolev space methods and considering various boundary conditions.

Contribution

It introduces a method to establish local well-posedness for the water wave problem without surface tension, accommodating multiple boundary configurations and deriving pressure conditions mathematically.

Findings

Proves local existence under Rayleigh-Taylor condition.

Handles multiple boundary and gravity configurations.

Derives pressure equality as a consequence of weak solutions.

Abstract

We study the free boundary evolution between two irrotational, incompressible and inviscid fluids in 2-D without surface tension. We prove local-existence in Sobolev spaces when, initially, the difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition. The well-posedness of the full water wave problem was first obtained by Wu \cite{Wu}. The methods introduced in this paper allows us to consider multiple cases: with or without gravity, but also a closed boundary or a periodic boundary with the fluids placed above and below it. It is assumed that the initial interface does not touch itself, being a part of the evolution problem to check that such property prevails for a short time, as well as it does the Rayleigh-Taylor condition, depending conveniently upon the initial data. The addition of the pressure equality to the contour dynamic equations is obtained as a mathematical consequence, and not as a physical assumption, from the mere fact that we are dealing with weak solutions of Euler's equation in the whole space.