Boundary regularity of rotating vortex patches

2012/08/31 by Taoufik Hmidi, Joan Mateu, Joan Verdera · 1 citation
Mathematics · #math.AP #math.CA #msc:30E20 #msc:35Q35 #msc:76B03 #msc:76B47

paper · pdf · doi:10.1007/s00205-013-0618-8

published as Arch. Ration. Mech.Anal. 209(2013), 171-208 · Various proofs have been shortened. One added reference

arxiv created 2014/10/31 · arxiv updated 2015/06/11

Abstract

We show that the boundary of a rotating vortex patch (or V-state, in the terminology of Deem and Zabusky) is of class Cⁱnfinity provided the patch is close enough to the bifurcation circle in the Lipschitz norm. The rotating patch is convex if it is close enough to the bifurcation circle in the C² norm. Our proof is based on Burbea's approach to V-states. Thus conformal mapping plays a relevant role as well as estimating, on Hölder spaces, certain non-convolution singular integral operators of Calderón-Zygmund type.

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