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.
"Two-dimensional gravity water waves with constant vorticity, I: Cubic lifespan." Anal. PDE 12 (4) 903 - 967, 2019. https://doi.org/10.2140/apde.2019.12.903