The Slightly Supercritical Euler Equations: Smooth Solutions and Vortex Patches
Abstract
We investiage the (slightly) supercritical 2D Euler equations. The paper consists of two parts. In the first part we prove wellposedness 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 supercritical regime.This paper extends the results of Chae, Constantin, and Wu where they prove wellposedness for the socalled LogLogEuler equation. We also extend the classical results of Chemin and BertozziConstantin 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2013
 DOI:
 10.48550/arXiv.1308.1155
 arXiv:
 arXiv:1308.1155
 Bibcode:
 2013arXiv1308.1155E
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 34 pages