The Slightly Supercritical Euler Equations: Smooth Solutions and Vortex Patches

TL;DR

This paper studies the slightly super-critical 2-D Euler equations, establishing well-posedness in $C^s$ spaces and proving global regularity for vortex patches, extending classical results to a more challenging super-critical regime.

Contribution

It proves well-posedness in $C^s$ spaces for all $s>0$ and extends global regularity results for vortex patches to the super-critical case, building on prior foundational work.

Findings

Well-posedness in $C^s$ spaces for all $s>0$

Global regularity for vortex patches in super-critical regime

Extension of classical vortex patch results to super-critical case

Abstract

We investiage the (slightly) super-critical 2-D Euler equations. The paper consists of two parts. In the first part we prove well-posedness in $C^s$ spaces for all $s>0.$ We also give growth estimates for the $C^s$ norms of the vorticity for $0< s \leq 1.$ In the second part we prove global regularity for the vortex patch problem in the super-critical regime.This paper extends the results of Chae, Constantin, and Wu where they prove well-posedness for the so-called LogLog-Euler equation. We also extend the classical results of Chemin and Bertozzi-Constantin on the vortex patch problem to the slightly supercritical case. The supercritical vortex patch problem introduces several extra difficulties which are overcome via delicate estimates which take advantage of the extra tangential regularity of the vortex patches. Both problems we study are done in the setting of the whole space.