Confinement of vorticity for the 2D Euler-alpha equations

TL;DR

This paper proves that for the 2D Euler-alpha equations, the unfiltered vorticity remains confined within a disk whose radius grows very slowly over time, extending known results from classical Euler equations.

Contribution

It establishes a confinement result for vorticity support in the Euler-alpha equations, generalizing previous results from classical Euler equations to this regularized model.

Findings

Vorticity support remains within a growing disk over time.

The disk's radius grows no faster than (t log t)^{1/4}.

Results extend classical Euler confinement to Euler-alpha equations.

Abstract

In this article we consider weak solutions of the Euler-$\alpha$ equations in the full plane. We take, as initial unfiltered vorticity, an arbitrary nonnegative, compactly supported, bounded Radon measure. Global well-posedness for the corresponding initial value problem is due M. Oliver and S. Shkoller. We show that, for all time, the support of the unfiltered vorticity is contained in a disk whose radius grows no faster than $\mathcal{O}((t\log t)^{1/4})$. This result is an adaptation of the corresponding result for the incompressible 2D Euler equations with initial vorticity compactly supported, nonnegative, and $p$-th power integrable, $p>2$, due to D. Iftimie, T. Sideris and P. Gamblin and, independently, to Ph. Serfati.