Nonlinear stability of vortex patches
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- by Yun Tang PDF
- Trans. Amer. Math. Soc. 304 (1987), 617-638 Request permission
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.References
-
V. I. Arnold, Conditions for nonlinear stability of stationary plane curlinear flows of an ideal fluid, Soviet Math. Dokl. 6 (1965), 773-777.
—, On an a priori estimate in the theory of hydrodynamical stability, Amer. Math. Soc. Transl. 79 (1969), 267-269.
- V. Arnold, Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. MR 0474391
- 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) Lecture Notes in Math., vol. 503, Springer, Berlin, 1976, pp. 8–29. MR 0671099
- 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 G. S. Deem and N. J. Zabusky, Vortex waves: stationary ‘$V$-states’, interactions, recurrence, and breaking, Phys. Rev. Lett. 40 (1978), 859-862. D. G. Dritschel, The stability and energetics of co-rotating uniform vortics (preprint), 1984.
- 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 271984, DOI 10.2307/1970699 L. Kelvin (Sir W. Thomson), On the vibrations of a columnar vortex, Philos. Mag. 5 (1880), 155. H. Lamb, Hydrodynamics, Dover, New York, 1945. A. E. H. Love, On the stability of certain vortex motions, Proc. Roy. Soc. London 25 (1893), 18-42.
- 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, DOI 10.1016/0167-2789(83)90134-3 T. G. McKee, Existence and structure on non-circular stationary vortices, Thesis, Brown University, 1981. R. T. Pierrehumbert, A family of steady, translating vortex pairs with distributed vorticity, J. Fluid Mech. 99 (1980), 129-144. P. G. Saffman, Vortex interactions and coherent structures in turbulence, Transition and Turbulence (Ed., R. E. Meyer), Academic Press, 1981, pp. 149-166.
- 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, DOI 10.1080/03605308308820293 —On the evolution of a concentrated vortex in an ideal fluid (preprint), 1984.
- Y. H. Wan and M. Pulvirenti, Nonlinear stability of circular vortex patches, Comm. Math. Phys. 99 (1985), no. 3, 435–450. MR 795112
Additional Information
- © Copyright 1987 American Mathematical Society
- 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