A min-max variational approach to the existence of gravity water waves

TL;DR

This paper introduces a variational method using a mountain pass theorem to prove the existence of gravity water waves in two dimensions, without symmetry or connectivity assumptions, broadening the scope of fluid equilibrium analysis.

Contribution

It develops a novel variational framework for gravity water waves that does not depend on symmetry, connectivity, or monotonicity, and applies to Bernoulli-type free boundary problems.

Findings

Proves existence of gravity water waves using a min-max variational approach.

Framework applicable to various fluid equilibrium problems.

Constructs solutions to Bernoulli free boundary problems.

Abstract

We establish the existence of gravity water waves by applying a mountain pass theorem to a singular perturbation of the Alt-Caffarelli functional associated with the two-dimensional water wave equations. Our approach is formulated entirely in physical coordinates and does not require the air phase to be connected, nor does it rely on symmetry or monotonicity in the $x$ or $y$ directions. The framework presented allows for both a variational approach to a variety of fluid equilibrium problems and for construction of min-max solutions to Bernoulli-type free boundary problems.