Nonlinear stability of vortex patches
Author:
Yun Tang
Journal:
Trans. Amer. Math. Soc. 304 (1987), 617638
MSC:
Primary 76C05; Secondary 35B35, 35Q10
MathSciNet review:
911087
Fulltext PDF Free Access
Abstract 
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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 twodimensional 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.
 [1]
V. I. Arnold, Conditions for nonlinear stability of stationary plane curlinear flows of an ideal fluid, Soviet Math. Dokl. 6 (1965), 773777.
 [2]
, On an a priori estimate in the theory of hydrodynamical stability, Amer. Math. Soc. Transl. 79 (1969), 267269.
 [3]
V.
I. Arnold, Mathematical methods of classical mechanics,
SpringerVerlag, New YorkHeidelberg, 1978. Translated from the Russian by
K. Vogtmann and A. Weinstein; Graduate Texts in Mathematics, 60. MR 0690288
(57 #14033b)
 [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
(58 #32375)
 [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
(85e:76015)
 [6]
G. S. Deem and N. J. Zabusky, Vortex waves: stationary `states', interactions, recurrence, and breaking, Phys. Rev. Lett. 40 (1978), 859862.
 [7]
D. G. Dritschel, The stability and energetics of corotating 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
(42 #6865)
 [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), 1842.
 [12]
Jerrold
Marsden and Alan
Weinstein, Coadjoint orbits, vortices, and Clebsch variables for
incompressible fluids, Phys. D 7 (1983),
no. 13, 305–323. Order in chaos (Los Alamos, N.M., 1982). MR 719058
(85g:58039), http://dx.doi.org/10.1016/01672789(83)901343
 [13]
T. G. McKee, Existence and structure on noncircular 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), 129144.
 [15]
P. G. Saffman, Vortex interactions and coherent structures in turbulence, Transition and Turbulence (Ed., R. E. Meyer), Academic Press, 1981, pp. 149166.
 [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
(85g:35110), http://dx.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
(86i:76028)
 [1]
 V. I. Arnold, Conditions for nonlinear stability of stationary plane curlinear flows of an ideal fluid, Soviet Math. Dokl. 6 (1965), 773777.
 [2]
 , On an a priori estimate in the theory of hydrodynamical stability, Amer. Math. Soc. Transl. 79 (1969), 267269.
 [3]
 , Mathematical methods of classical mechanics, Graduate Texts in Math. #60, Springer, New York, 1978. MR 0690288 (57:14033b)
 [4]
 T. B. Benjamin, The alliance of practical and analytic insights into the nonlinear problems of fluid mechanics, Applications of Methods of Functional Analysis to Problems of Mechanics, Lecture Notes in Math., vol. 503, SpringerVerlag, 1976, pp. 829. MR 0671099 (58:32375)
 [5]
 J. Burbea, Vortex motion and their stability, Proc. Nonlinear Phenomena in Math. Sci. (Arlington), Academic Press, 1982, pp. 147158. MR 727976 (85e:76015)
 [6]
 G. S. Deem and N. J. Zabusky, Vortex waves: stationary `states', interactions, recurrence, and breaking, Phys. Rev. Lett. 40 (1978), 859862.
 [7]
 D. G. Dritschel, The stability and energetics of corotating uniform vortics (preprint), 1984.
 [8]
 E. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 (1970), 102163. MR 0271984 (42:6865)
 [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), 1842.
 [12]
 J. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D 7 (1983), 305323. MR 719058 (85g:58039)
 [13]
 T. G. McKee, Existence and structure on noncircular 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), 129144.
 [15]
 P. G. Saffman, Vortex interactions and coherent structures in turbulence, Transition and Turbulence (Ed., R. E. Meyer), Academic Press, 1981, pp. 149166.
 [16]
 B. Turkington, On steady vortex flow in two dimensions. I, Comm. Partial Differential Equations 8 (1983), 9991030. MR 702729 (85g:35110)
 [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), 435450. MR 795112 (86i:76028)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719870911087X
PII:
S 00029947(1987)0911087X
Article copyright:
© Copyright 1987
American Mathematical Society
