Boundary regularity of rotating vortex patches

Taoufik Hmidi (1) , Joan Mateu (2) , Joan Verdera (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 UAB - Departament de Matemàtiques [Barcelona]

We show that the boundary of a rotating vortex patch (or V-state, in the terminology of Deem and Zabusky) is of class C^infinity 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 C^2 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.

Domaines: Analyse classique [math.CA]; Equations aux dérivées partielles [math.AP]

Taoufik Hmidi, Joan Mateu, Joan Verdera. Boundary regularity of rotating vortex patches. Archive for Rational Mechanics and Analysis, 2013, 209 (1), pp.171-208. ⟨10.1007/s00205-013-0618-8⟩. ⟨hal-00727718⟩