Advertisement

Archive for Rational Mechanics and Analysis

, Volume 209, Issue 1, pp 171–208 | Cite as

Boundary Regularity of Rotating Vortex Patches

  • Taoufik Hmidi
  • Joan Mateu
  • Joan Verdera
Article

Abstract

We show that the boundary of a rotating vortex patch (or V-state, in the terminology of Deem and Zabusky) is C , provided the patch is close to the bifurcation circle in the Lipschitz norm. The rotating patch is also convex if it is close to the bifurcation circle in the C 2 norm. Our proof is based on Burbea’s approach to V-states.

Keywords

Vorticity Unit Disc Conformal Mapping Jordan Curve Open Unit Disc 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aref H.: Integrable, chaotic, and turbulent vortex motion in two-dimensional flows. Annu. Rev. Fluid Mech. 15, 345–389 (1983)MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Bertozzi, A.L., Majda, A.J.: Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002Google Scholar
  3. 3.
    Burbea J.:: Motions of vortex patches. Lett. Math. Phys. 6, 1–16 (1982)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Crandall M.G., Rabinowitz P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Deem G.S., Zabusky N.J.: Vortex waves: stationary “V-states”, Interactions, Recurrence, and Breaking. Phys. Rev. Lett. 40(13), 859–862 (1978)ADSCrossRefGoogle Scholar
  6. 6.
    Duren, P.L., Univalent Functions, Grundlehren der mathematischen Wissenschaften, Vol. 259, Springer-Verlag, New York, 1983Google Scholar
  7. 7.
    Gatto A.E.: On the boundedness on inhomogeneous Lipschitz spaces of fractional integrals, singular integrals and hypersingular integrals associated to non-doubling measures on metric spaces. Collect. Math. 60, 101–114 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Mateu J., Orobitg J., Verdera J.: Extra cancellation of even Calderón–Zygmund operators and quasiconformal mappings. J. Math. Pures Appl. 91(4), 402–431 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Lamb, H.: Hydrodynamics, Dover Publications, New York, 1945Google Scholar
  10. 10.
    Pommerenke, Ch.: Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992Google Scholar
  11. 11.
    Verdera J.: L 2 boundedness of the Cauchy integral and Menger curvature. Contemp. Math. 277, 139–158 (2001)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Warschawski S.E.: On the higher derivatives at the boundary in conformal mapping. Trans. Am. Math. Soc. 38(2), 310–340 (1935)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Wittmann R.: Application of a Theorem of M.G. Krein to singular integrals. Trans. Am. Math. Soc. 299(2), 581–599 (1987)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Wu H.M., Overman E.A. II, Zabusky N.J.: Steady-state solutions of the Euler equations in two dimensions: rotating and translating V-states with limiting cases I. lgorithms and results. J. Comput. Phys. 53, 42–71 (1984)MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.IRMARUniversité de Rennes 1Rennes CedexFrance
  2. 2.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra, BarcelonaSpain

Personalised recommendations