Vortex rings: existence and asymptotic estimates

Friedman, Avner; Turkington, Bruce

doi:10.1090/S0002-9947-1981-0628444-6

Vortex rings: existence and asymptotic estimates
HTML articles powered by AMS MathViewer

by Avner Friedman and Bruce Turkington PDF

Trans. Amer. Math. Soc. 268 (1981), 1-37 Request permission

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.

References

J. F. G. Auchmuty, Existence of axisymmetric equilibrium figures, Arch. Rational Mech. Anal. 65 (1977), no. 3, 249–261. MR 446076, DOI 10.1007/BF00280443
J. F. G. Auchmuty and Richard Beals, Variational solutions of some nonlinear free boundary problems, Arch. Rational Mech. Anal. 43 (1971), 255–271. MR 337260, DOI 10.1007/BF00250465
G. K. Batchelor, An introduction to fluid dynamics, Second paperback edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1999. MR 1744638
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

Quelques questions liées à la théorie des tourbillons stationnaires dans un fluid ideal

M. S. Berger and L. E. Fraenkel, On nonlinear desingularization, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 1, 165–167. MR 551754, DOI 10.1090/S0273-0979-1980-14704-7
M. S. Berger and L. E. Fraenkel, Nonlinear desingularization in certain free-boundary problems, Comm. Math. Phys. 77 (1980), no. 2, 149–172. MR 589430
Paul F. Byrd and Morris D. Friedman, Handbook of elliptic integrals for engineers and scientists, Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York-Heidelberg, 1971. Second edition, revised. MR 0277773
Luis A. Caffarelli and Avner Friedman, The shape of axisymmetric rotating fluid, J. Functional Analysis 35 (1980), no. 1, 109–142. MR 560219, DOI 10.1016/0022-1236(80)90082-8
Luis A. Caffarelli and Avner Friedman, Asymptotic estimates for the plasma problem, Duke Math. J. 47 (1980), no. 3, 705–742. MR 587175

On steady vortex rings of small cross-section in an ideal fluid

316

Examples of steady vortex rings of small cross-section in an ideal fluid

51

L. E. Fraenkel and M. S. Berger, A global theory of steady vortex rings in an ideal fluid, Acta Math. 132 (1974), 13–51. MR 422916, DOI 10.1007/BF02392107
Avner Friedman and Bruce Turkington, Asymptotic estimates for an axisymmetric rotating fluid, J. Functional Analysis 37 (1980), no. 2, 136–163. MR 578929, DOI 10.1016/0022-1236(80)90038-5
Avner Friedman and Bruce Turkington, The oblateness of an axisymmetric rotating fluid, Indiana Univ. Math. J. 29 (1980), no. 5, 777–792. MR 589442, DOI 10.1512/iumj.1980.29.29056
Avner Friedman and Bruce Turkington, Existence and dimensions of a rotating white dwarf, J. Differential Equations 42 (1981), no. 3, 414–437. MR 639231, DOI 10.1016/0022-0396(81)90114-5

Über Integrate der hydrodynamischen Gleichungen, welche den Wirbetwegungen entsprechen

55

On a spherical vortex

185

Hydrodynamics

Wei Ming Ni, On the existence of global vortex rings, J. Analyse Math. 37 (1980), 208–247. MR 583638, DOI 10.1007/BF02797686