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Communications in Mathematical Physics

, Volume 337, Issue 1, pp 321–377 | Cite as

On the V-states for the Generalized Quasi-Geostrophic Equations

  • Zineb Hassainia
  • Taoufik Hmidi
Article

Abstract

We prove the existence of the V-states for the generalized inviscid SQG equations with \({\alpha \in ]0, 1[.}\) These structures are special rotating simply connected patches with m-fold symmetry bifurcating from the trivial solution at some explicit values of the angular velocity. This produces, inter alia, an infinite family of non stationary global solutions with uniqueness.

Keywords

Vortex Angular Velocity Euler Equation Conformal Mapping Trivial Solution 
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.

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References

  1. 1.
    Ambaum M.H.P., Harvey B.J.: Perturbed Rankine vortices in surface quasi-geostrophic dynamics. Geophys. Astrophys. Fluid Dyn. 105(4–5), 377–391 (2011)ADSMathSciNetGoogle Scholar
  2. 2.
    Aref H.: Integrable, chaotic, and turbulent vortex motion in two-dimensional flows. Annu. Rev. Fluid Mech. 15, 345–389 (1983)CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Bertozzi A.L., Majda A.J.: Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)Google Scholar
  4. 4.
    Burbea J.: Motions of vortex patches. Lett. Math. Phys. 6, 1–16 (1982)CrossRefADSzbMATHMathSciNetGoogle Scholar
  5. 5.
    Castro A., Córdoba D., Gómez-Serrano J., Martín Zamora A.: Remarks on geometric properties of SQG sharp fronts and α-patches. Discrete Contin. Dyn. Syst. 34(12), 5045–5059 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Chae D., Constantin P., Córdoba D., Gancedo F., Wu J.: Generalized surface quasi-geostrophic equations with singular velocities. Commun. Pure Appl. Math. 65(8), 1037–1066 (2012)CrossRefzbMATHGoogle Scholar
  7. 7.
    Chaplygin, S.A.: On a pulsating cylindrical vortex. Translated from the 1899 Russian original by G. Krichevets, edited by D. Blackmore and with comments by V. V. Meleshko. Regul. Chaotic Dyn. 12(1), 101–116 (2007)Google Scholar
  8. 8.
    Chemin, J.Y.: Fluides parfaits incompressibles. Astérisque, vol. 230, Société Mathématique de France (1995)Google Scholar
  9. 9.
    Constantin P., Majda A.J., Tabak E.: Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7(6), 1495–1533 (1994)CrossRefADSzbMATHMathSciNetGoogle Scholar
  10. 10.
    Córdoba D., Fontelos M.A., Mancho A.M., Rodrigo J.L.: Evidence of singularities for a family of contour dynamics equations. Proc. Natl. Acad. Sci. USA 102, 5949–5952 (2005)CrossRefADSzbMATHMathSciNetGoogle Scholar
  11. 11.
    Crandall M.G., Rabinowitz P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Deem G.S., Zabusky N.J.: Vortex waves: stationary “V-states”, interactions, recurrence, and breaking. Phys. Rev. Lett. 40(13), 859–862 (1978)CrossRefADSGoogle Scholar
  13. 13.
    Duren P.L.: Univalent functions, Grundlehren der mathematischen Wissenschaften, vol. 259. Springer, New York (1983)Google Scholar
  14. 14.
    Flierl G.R., Polvani L.M.: Generalized Kirchhoff vortices. Phys. Fluids 29, 2376–2379 (1986)CrossRefADSzbMATHGoogle Scholar
  15. 15.
    Gancedo F.: Existence for the α-patch model and the QG sharp front in Sobolev spaces. Adv. Math. 217(6), 2569–2598 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Harvey B.J., Ambaum M.H.P.: Perturbed Rankine vortices in surface quasi-geostrophic dynamics. Geophys. Astrophys. Fluid Dyn. 105(4–5), 377–391 (2011)CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Held I., Pierrehumbert R., Garner S., Swanson K.: Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20 (1995)CrossRefADSzbMATHMathSciNetGoogle Scholar
  18. 18.
    Hmidi T., Mateu J., Verdera J.: Boundary regularity of rotating vortex patches. Arch. Ration. Mech. Anal. 209(1), 171–208 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Hmidi, T., Mateu, J., Verdera, J.: On rotating doubly connected vortices. J. Differ. Equations 258(4), 1395–1429 (2015)Google Scholar
  20. 20.
    Hmidi, T.: On the trivial solutions for the rotating patch model. arXiv:1409.8469
  21. 21.
    Juckes, M.: Quasigeostrophic dynamics of the tropopause. J. Armos. Sci., 2756–2768 (1994)Google Scholar
  22. 22.
    Kida S.: Motion of an elliptical vortex in a uniform shear flow. J. Phys. Soc. Japan 50, 3517–3520 (1981)CrossRefADSGoogle Scholar
  23. 23.
    Kielhöfer H.: Bifurcation Theory: An Introduction With Applications to Partial Differential Equations. Springer, Berlin (2011)Google Scholar
  24. 24.
    Kirchhoff, G.: Vorlesungen uber mathematische Physik, Leipzig (1874)Google Scholar
  25. 25.
    Lamb H.: Hydrodynamics. Dover Publications, New York (1945)Google Scholar
  26. 26.
    Lapeyre G., Klein P.: Dynamics of the upper oceanic layers in terms of surface quasigeostrophic theory. J. Phys. Oceanogr. 36, 165–176 (2006)CrossRefADSMathSciNetGoogle Scholar
  27. 27.
    Magnus W., Oberhettinger F.: Formeln und satze fur die speziellen funktionen der mathematischen physik. Springer, Berlin (1948)CrossRefGoogle Scholar
  28. 28.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Neu J.: The dynamics of columnar vortex in an imposed strain. Phys. Fluids 27, 2397–2402 (1984)CrossRefADSzbMATHGoogle Scholar
  30. 30.
    Newton P.K.: The N-Vortex Problem. Analytical Techniques. Springer, New York (2001)zbMATHGoogle Scholar
  31. 31.
    Pommerenke Ch.: Boundary Behaviour of Conformal Maps. Springer, Berlin (1992)zbMATHGoogle Scholar
  32. 32.
    Rodrigo J.L.: On the evolution of sharp fronts for the quasi-geostrophic equation. Commun. Pure Appl. Math. 58(6), 821–866 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Verdera J.: L 2 boundedness of the Cauchy integral and Menger curvature. Contemp. Math. 277, 139–158 (2001)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Warschawski S.E.: On the higher derivatives at the boundary in conformal mapping. Trans. Am. Math. Soc. 38(2), 310–340 (1935)CrossRefMathSciNetGoogle Scholar
  35. 35.
    Watson G.A.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1944)zbMATHGoogle Scholar
  36. 36.
    Wittmann R.: Application of a theorem of M.G. Krein to singular integrals. Trans. Am. Math. Soc. 299(2), 581–599 (1987)zbMATHMathSciNetGoogle Scholar
  37. 37.
    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. Algorithms ans results. J. Comput. Phys. 53, 42–71 (1984)CrossRefADSzbMATHMathSciNetGoogle Scholar
  38. 38.
    Yudovich Y.: Nonstationary flow of an ideal incompressible liquid. Zh. Vych. Mat. 3, 1032–1066 (1963)zbMATHGoogle Scholar
  39. 39.
    Zabusky N., Hughes M.H., Roberts K.V.: Contour dynamics for the Euler equations in two dimensions. J. Comput. Phys. 30(1), 96–106 (1979)CrossRefADSzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.IRMAR, Université de Rennes 1Rennes CedexFrance

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