Two dimensional gravity water waves with constant vorticity: I. Cubic lifespan

TL;DR

This paper studies the two-dimensional gravity water wave equations with constant vorticity, establishing local well-posedness for large data and cubic lifespan bounds for small data solutions in a holomorphic coordinate framework.

Contribution

It introduces a novel analysis in holomorphic coordinates and proves well-posedness and lifespan bounds for water waves with vorticity, extending previous results.

Findings

Proved local well-posedness for large initial data.

Established cubic lifespan bounds for small data solutions.

Utilized holomorphic coordinate techniques for analysis.

Abstract

This article is concerned with the incompressible, infinite depth water wave equation in two space dimensions, with gravity and constant vorticity but with no surface tension. We consider this problem expressed in position-velocity potential holomorphic coordinates, and prove local well-posedness for large data, as well as cubic lifespan bounds for small data solutions.