The water waves equations: from Zakharov to Euler

TL;DR

This paper demonstrates how solutions to the classical Euler equations for water waves can be derived from the Zakharov/Craig-Sulem formulation, even in rough domains with minimal regularity assumptions.

Contribution

It establishes a rigorous link between the Zakharov formulation and classical Euler solutions in low-regularity settings.

Findings

Pressure term can be defined from Zakharov formulation

Solutions exist in rough domains with minimal Sobolev regularity

Euler solutions are obtained from water wave equations

Abstract

Starting form the Zakharov/Craig-Sulem formulation of the water-waves equations, we prove that one can define a pressure term and hence obtain a solution of the classical Euler equations. It is proved that these results hold in rough domains, under minimal assumptions on the regularity to ensure, in terms of Sobolev spaces, that the solutions are $C^1$.