Regularity of the velocity field for Euler vortex patch evolution

TL;DR

This paper investigates the regularity properties of velocity fields in vortex patch problems for 2-D and 3-D incompressible Euler equations, establishing conditions under which the velocity and boundary regularities are preserved over time.

Contribution

It proves the preservation of velocity field regularity in 2-D and 3-D vortex patches with specific Sobolev-class boundary regularities, extending understanding of vortex dynamics.

Findings

Velocity field on both sides of the boundary remains $H^k$ regular in 2-D.

Existence of solutions with boundary in $H^{k-0.5}$ in 3-D.

Velocity fields maintain $H^k$ regularity over finite time intervals.

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, $k \ge 4$, 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 $k \ge 3$.