Exponential Vorticity Hessian Growth in Capillary Liquid Drop in Two Dimensions

TL;DR

This paper demonstrates that in a two-dimensional ideal fluid droplet with surface tension, the vorticity Hessian can grow exponentially over time, even from small initial velocities.

Contribution

It constructs specific initial conditions showing exponential growth of the vorticity Hessian in a 2D droplet with surface tension, revealing new dynamical behavior.

Findings

Vorticity Hessian grows exponentially over time.

Small initial velocities can lead to unbounded vorticity growth.

Surface tension influences the evolution of vorticity Hessian.

Abstract

In this work, we concern ourselves with the evolution of a droplet of an ideal fluid in two dimensions, which has nontrivial bulk vorticity and is only subject to the effects of surface tension. We construct initial data with initial domain being a disk and initial velocity being arbitrarily small, such that the vorticity Hessian grows exponentially infinitely in time.