Small scale creation for 2D free boundary Euler equations with surface tension

TL;DR

This paper investigates the growth of vorticity gradients in 2D free boundary Euler equations with surface tension, extending known results to more complex fluid domain deformations and boundary conditions.

Contribution

It generalizes the double-exponential vorticity gradient growth result to free boundary flows, overcoming challenges posed by domain deformation and boundary dynamics.

Findings

Vorticity gradient grows at least double-exponentially over time.

Constructed initial data with flat boundary and small velocity.

Extended the Kiselev–Sverák result to free boundary scenarios.

Abstract

In this paper, we study the 2D free boundary incompressible Euler equations with surface tension, where the fluid domain is periodic in $x_1$, and has finite depth. We construct initial data with a flat free boundary and arbitrarily small velocity, such that the gradient of vorticity grows at least double-exponentially for all times during the lifespan of the associated solution. This work generalizes the celebrated result by Kiselev--{\v{S}}ver{\'a}k to the free boundary setting. The free boundary introduces some major challenges in the proof due to the deformation of the fluid domain and the fact that the velocity field cannot be reconstructed from the vorticity using the Biot-Savart law. We overcome these issues by deriving uniform-in-time control on the free boundary and obtaining pointwise estimates on an approximate Biot-Savart law.