Regularity of the velocity field for Euler vortex patch evolution

2015/09/25 by Coutand, Daniel, Shkoller, Steve
#35J57 #76B03 #76B47 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1509.07778

Abstract

We consider the vortex patch problem for both the 2-D and 3-D incompressible Euler equations. In 2-D, we prove that for vortex patches with H^k-0.5 Sobolev-class contour regularity, k ≥ 4, the velocity field on both sides of the vortex patch boundary has H^k regularity for all time. In 3-D, we establish existence of solutions to the vortex patch problem on a finite-time interval [0,T], and we simultaneously establish the H^k-0.5 regularity of the two-dimensional vortex patch boundary, as well as the H^k regularity of the velocity fields on both sides of vortex patch boundary, for k ≥ 3.

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