Equivalence of weak formulations of the steady water waves equations

TL;DR

This paper demonstrates that three different weak formulations of the steady water waves equations are mathematically equivalent under certain regularity conditions, unifying various approaches in the study of water waves.

Contribution

It proves the equivalence of velocity, stream function, and Dubreil-Jacotin formulations for steady water waves under weak regularity assumptions.

Findings

The three formulations are mathematically equivalent.

Weak Holder regularity suffices for the equivalence.

Unifies different mathematical approaches to water wave problems.

Abstract

We prove the equivalence of three weak formulations of the steady water waves equations, namely the velocity formulation, the stream function formulation, and the Dubreil-Jacotin formulation, under weak Holder regularity assumptions on their solutions.