Regularity of the velocity field for Euler vortex patch evolution

Coutand, Daniel; Shkoller, Steve

doi:10.1090/tran/7058

Regularity of the velocity field for Euler vortex patch evolution
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by Daniel Coutand and Steve Shkoller PDF

Trans. Amer. Math. Soc. 370 (2018), 3689-3720 Request permission

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 $k \ge 3$.

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