Fast growth of the vorticity gradient in symmetric smooth domains for 2D incompressible ideal flow

TL;DR

This paper demonstrates that in symmetric smooth domains, the vorticity gradient for 2D ideal flow can grow double exponentially over time, highlighting extreme vorticity amplification.

Contribution

It constructs specific initial data showing double exponential growth of vorticity gradient in symmetric smooth domains, extending previous theoretical results.

Findings

Vorticity gradient grows double exponentially in time.

Construction of initial data leading to unbounded growth.

Extension of Kiselev and Sverák's results to bounded smooth domains.

Abstract

We construct an initial data for the two-dimensional Euler equation in a bounded smooth symmetric domain such that the gradient of vorticity in $L^{\infty}$ grows as a double exponential in time for all time. Our construction is based on the recent result by Kiselev and \v{S}ver\'{a}k.