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Transactions of the American Mathematical Society

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Vortex rings: existence and asymptotic estimates

Authors: Avner Friedman and Bruce Turkington
Journal: Trans. Amer. Math. Soc. 268 (1981), 1-37
MSC: Primary 76C05; Secondary 49H05
MathSciNet review: 628444
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Abstract: The existence of a family of steady vortex rings is established by a variational principle. Further, the asymptotic behavior of the solutions is obtained for limiting values of an appropriate parameter $ \lambda $; as $ \lambda \to \infty $ the vortex ring tends to a torus whose cross-section is an infinitesimal disc.

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  • [1] J. F. G. Auchmuty, Existence of axisymmetric equilibrium figures, Arch. Rational Mech. Anal. 65 (1977), 249-261. MR 0446076 (56:4408)
  • [2] J. F. G. Auchmuty and R. Beals, Variational solutions of some nonlinear free boundary problems, Arch. Rational Mech. Anal. 43 (1971), 255-271. MR 0337260 (49:2029)
  • [3] G. K. Batchelor, An introduction to fluid dynamics, Cambridge Univ. Press, New York, 1974. MR 1744638 (2000j:76001)
  • [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, Springer-Verlag, Berlin, 1976, pp. 8-29. MR 0671099 (58:32375)
  • [5] H. Berestycki, Quelques questions liées à la théorie des tourbillons stationnaires dans un fluid ideal (to appear).
  • [6] M. S. Berger and L. E. Fraenkel, On nonlinear desingularization, Bull. Amer. Math. Soc. (N.S.) 2 (1980), 165-167. MR 551754 (81b:58012)
  • [7] -, Nonlinear desingularization in certain free boundary problems, Comm. Math. Phys. (to appear). MR 589430 (81m:35055)
  • [8] P. E. Byrd and M. D. Friedman, Handbook of elliptic integrals for engineers and scientists, Springer-Verlag, Berlin, 1971. MR 0277773 (43:3506)
  • [9] L. A. Caffarelli and A. Friedman, The shape of axisymmetric rotating fluid, J. Funct. Anal. 35 (1980), 109-142. MR 560219 (81j:76090)
  • [10] -, Asymptotic estimates for the plasma problem, Duke Math. J. 47 (1980), 705-742. MR 587175 (82b:35048)
  • [11] L. E. Fraenkel, On steady vortex rings of small cross-section in an ideal fluid, Proc. Roy. Soc. London Ser. A 316 (1970), 29-62.
  • [12] -, Examples of steady vortex rings of small cross-section in an ideal fluid, J. Fluid Mech. 51 (1972), 119-135.
  • [13] L. E. Fraenkel and M. S. Berger, A global theory of steady vortex rings in an ideal fluid, Acta Math. 132 (1974), 14-51. MR 0422916 (54:10901)
  • [14] A. Friedman and B. Turkington, Asymptotic estimate for an axisymmetric rotating fluid, J. Funct. Anal. 37 (1980), 136-163. MR 578929 (81h:76058)
  • [15] -, The oblateness of an axisymmetric rotating fluid, Indiana Univ. Math. J. 29 (1980), 777-792. MR 589442 (82c:76104)
  • [16] -, Existence and dimensions of a rotating white dwarf, J. Differential Equations (to appear). MR 639231 (83a:85002)
  • [17] H. Helmholtz, Über Integrate der hydrodynamischen Gleichungen, welche den Wirbetwegungen entsprechen, J. Reine Angew. Math. 55 (1858), 25-55.
  • [18] M. J. M. Hill, On a spherical vortex, Philos. Trans. Roy. Soc. London Ser. A 185 (1894), 213-245.
  • [19] H. Lamb, Hydrodynamics, Cambridge, 1932.
  • [20] W. M. Ni, On the existence of global vortex rings (to appear). MR 583638 (81i:76017)

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