Long-time Confinement near Special Vortex Crystals

TL;DR

This paper demonstrates the long-time confinement of solutions to 2D Euler equations with vorticity near special vortex crystals, highlighting stability properties of regular polygons with or without a central vortex.

Contribution

It introduces a method to control the support growth of solutions near vortex crystals, emphasizing stability in specific polygonal configurations.

Findings

Support of solutions remains confined over long times.

Regular polygons with a central vortex exhibit strong stability.

Long-time confinement is achieved for specific vortex arrangements.

Abstract

In this paper, we control the growth of the support of particular solutions to the Euler two-dimensional equations, whose vorticity is concentrated near special vortex crystals. These vortex crystals belong to the classical family of regular polygons with a central vortex, where we choose a particular intensity for the central vortex to have strong stability properties. A special case is the regular pentagon with no central vortex which also satisfies the stability properties required for the long-time confinement to work.