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

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 $L^{\mathrm{\infty }}$ 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.