The Slightly Supercritical Euler Equations: Smooth Solutions and Vortex\n Patches

2013/08/05 by Tarek M. Elgindi, Elgindi, Tarek M
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1308.1155

openalex publication_date 2013/08/05 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28

Abstract

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

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