Regularity of the velocity field for Euler vortex patch evolution

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 H^{k-0.5} Sobolev-class contour regularity, k ≥ 4 k \ge 4 , the velocity field on both sides of the vortex patch boundary has H k 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 ] [0,T] , and we simultaneously establish the H k − 0.5 H^{k-0.5} regularity of the two-dimensional vortex patch boundary, as well as the H k H^k regularity of the velocity fields on both sides of vortex patch boundary, for k ≥ 3 k \ge 3 .