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Transactions of the American Mathematical Society

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Nonlinear stability of vortex patches


Author: Yun Tang
Journal: Trans. Amer. Math. Soc. 304 (1987), 617-638
MSC: Primary 76C05; Secondary 35B35, 35Q10
DOI: https://doi.org/10.1090/S0002-9947-1987-0911087-X
MathSciNet review: 911087
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Abstract: To establish the nonlinear (Liapunov) stability of both circular and elliptical vortex patches in the plane for the nonlinear dynamical system generated by the two-dimensional Euler equations of incompressible, inviscid hydrodynamics. This is accomplished by using a relative variational principle in terms of energy function. A counterexample shows that our result in the case of an elliptical vortex patch is the best one that can be attained by applying the energy estimate.


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  • [1] V. I. Arnold, Conditions for nonlinear stability of stationary plane curlinear flows of an ideal fluid, Soviet Math. Dokl. 6 (1965), 773-777.
  • [2] -, On an a priori estimate in the theory of hydrodynamical stability, Amer. Math. Soc. Transl. 79 (1969), 267-269.
  • [3] V. Arnold, Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. MR 0474391
    V. I. Arnol′d, Mathematical methods of classical mechanics, Springer-Verlag, New York-Heidelberg, 1978. Translated from the Russian by K. Vogtmann and A. Weinstein; Graduate Texts in Mathematics, 60. MR 0690288
  • [4] T. Brooke Benjamin, The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics, Applications of methods of functional analysis to problems in mechanics (Joint Sympos., IUTAM/IMU, Marseille, 1975) Springer, Berlin, 1976, pp. 8–29. Lecture Notes in Math., 503. MR 0671099
  • [5] Jacob Burbea, Vortex motions and their stability, Nonlinear phenomena in mathematical sciences (Arlington, Tex., 1980) Academic Press, New York, 1982, pp. 147–158. MR 727976
  • [6] G. S. Deem and N. J. Zabusky, Vortex waves: stationary `$ V$-states', interactions, recurrence, and breaking, Phys. Rev. Lett. 40 (1978), 859-862.
  • [7] D. G. Dritschel, The stability and energetics of co-rotating uniform vortics (preprint), 1984.
  • [8] David G. Ebin and Jerrold Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid., Ann. of Math. (2) 92 (1970), 102–163. MR 0271984, https://doi.org/10.2307/1970699
  • [9] L. Kelvin (Sir W. Thomson), On the vibrations of a columnar vortex, Philos. Mag. 5 (1880), 155.
  • [10] H. Lamb, Hydrodynamics, Dover, New York, 1945.
  • [11] A. E. H. Love, On the stability of certain vortex motions, Proc. Roy. Soc. London 25 (1893), 18-42.
  • [12] Jerrold Marsden and Alan Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D 7 (1983), no. 1-3, 305–323. Order in chaos (Los Alamos, N.M., 1982). MR 719058, https://doi.org/10.1016/0167-2789(83)90134-3
  • [13] T. G. McKee, Existence and structure on non-circular stationary vortices, Thesis, Brown University, 1981.
  • [14] R. T. Pierrehumbert, A family of steady, translating vortex pairs with distributed vorticity, J. Fluid Mech. 99 (1980), 129-144.
  • [15] P. G. Saffman, Vortex interactions and coherent structures in turbulence, Transition and Turbulence (Ed., R. E. Meyer), Academic Press, 1981, pp. 149-166.
  • [16] Bruce Turkington, On steady vortex flow in two dimensions. I, II, Comm. Partial Differential Equations 8 (1983), no. 9, 999–1030, 1031–1071. MR 702729, https://doi.org/10.1080/03605308308820293
  • [17] -On the evolution of a concentrated vortex in an ideal fluid (preprint), 1984.
  • [18] Y. H. Wan and M. Pulvirenti, Nonlinear stability of circular vortex patches, Comm. Math. Phys. 99 (1985), no. 3, 435–450. MR 795112

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1987-0911087-X
Article copyright: © Copyright 1987 American Mathematical Society