On the growth of the support of positive vorticity for 2D Euler equation in an infinite cylinder

TL;DR

This paper investigates the long-term growth of the support of positive vorticity in the 2D Euler equations on an infinite cylinder, establishing an upper bound on its diameter growth rate over time.

Contribution

It provides a novel upper bound on the support's diameter growth for 2D Euler flows with positive vorticity in an infinite cylinder.

Findings

Support diameter grows at most like t^{1/3} log^2 t as t approaches infinity.

The result applies to initial vorticity that is non-negative, bounded, and compactly supported.

The study advances understanding of vorticity support dynamics in unbounded domains.

Abstract

We consider the incompressible 2D Euler equation in an infinite cylinder $\mathbb{R}\times \mathbb{T}$ in the case when the initial vorticity is non-negative, bounded, and compactly supported. We study $d(t)$, the diameter of the support of vorticity, and prove that it allows the following bound: $d(t)\leq Ct^{1/3}\log^2 t$ when $t\rightarrow\infty$.