Small scale creation in 2D gravity-capillary water waves with vorticity

TL;DR

This paper constructs specific initial conditions for 2D water waves with vorticity, demonstrating double-exponential growth in vorticity gradient, extending prior results to free-surface and unbounded domain settings.

Contribution

It generalizes previous vorticity growth results to free-surface water waves in unbounded domains, showing potential for extreme vorticity amplification.

Findings

Double-exponential growth of vorticity gradient demonstrated

Extension of Zlatos' result to free-surface setting

Extension of Hu--Luo--Yao's result to unbounded domains

Abstract

In this paper, we consider 2D incompressible Euler equations in an unbounded domain with a free surface and a fixed bottom at finite depth. The fluid motion is under the influence of gravity and surface tension. We construct initial data with a flat free surface and small velocity, such that the $L^\infty$ norm of the vorticity gradient has at least a double-exponential growth rate within the lifespan of the corresponding solution. This work generalizes the result of Zlatos to the free-surface setting and Hu--Luo--Yao to the case of an unbounded domain.