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Friedman, Avner; Turkington, Bruce

Vortex rings: existence and asymptotic estimates. (English) Zbl 0497.76031

Trans. Am. Math. Soc. 268, 1-37 (1981).
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Cited in 43 Documents

MSC:

76B47 Vortex flows for incompressible inviscid fluids
76G25 General aerodynamics and subsonic flows
49S05 Variational principles of physics
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J20 Variational methods for second-order elliptic equations

Keywords:

capacity methods; steady vortex rings; kinetic energy as functional of vorticity; constraints

Citations:

Zbl 0369.76048; Zbl 0282.76014; Zbl 0366.76083; Zbl 0225.49013
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Full Text: DOI

References:

[1] J. F. G. Auchmuty, Existence of axisymmetric equilibrium figures, Arch. Rational Mech. Anal. 65 (1977), no. 3, 249 – 261. · Zbl 0366.76083 · doi:10.1007/BF00280443
[2] J. F. G. Auchmuty and Richard Beals, Variational solutions of some nonlinear free boundary problems, Arch. Rational Mech. Anal. 43 (1971), 255 – 271. · Zbl 0225.49013 · doi:10.1007/BF00250465
[3] G. K. Batchelor, An introduction to fluid dynamics, Second paperback edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1999. · Zbl 0958.76001
[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.
[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), no. 1, 165 – 167. · Zbl 0432.35015
[7] M. S. Berger and L. E. Fraenkel, Nonlinear desingularization in certain free-boundary problems, Comm. Math. Phys. 77 (1980), no. 2, 149 – 172. · Zbl 0454.35087
[8] 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. · Zbl 0213.16602
[9] Luis A. Caffarelli and Avner Friedman, The shape of axisymmetric rotating fluid, J. Funct. Anal. 35 (1980), no. 1, 109 – 142. · Zbl 0439.35068 · doi:10.1016/0022-1236(80)90082-8
[10] Luis A. Caffarelli and Avner Friedman, Asymptotic estimates for the plasma problem, Duke Math. J. 47 (1980), no. 3, 705 – 742. · Zbl 0466.35033
[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. · Zbl 0195.55101
[12] -, Examples of steady vortex rings of small cross-section in an ideal fluid, J. Fluid Mech. 51 (1972), 119-135. · Zbl 0231.76013
[13] L. E. Fraenkel and M. S. Berger, A global theory of steady vortex rings in an ideal fluid, Acta Math. 132 (1974), 13 – 51. · Zbl 0282.76014 · doi:10.1007/BF02392107
[14] Avner Friedman and Bruce Turkington, Asymptotic estimates for an axisymmetric rotating fluid, J. Funct. Anal. 37 (1980), no. 2, 136 – 163. · Zbl 0435.35014 · doi:10.1016/0022-1236(80)90038-5
[15] Avner Friedman and Bruce Turkington, The oblateness of an axisymmetric rotating fluid, Indiana Univ. Math. J. 29 (1980), no. 5, 777 – 792. · Zbl 0484.76116 · doi:10.1512/iumj.1980.29.29056
[16] Avner Friedman and Bruce Turkington, Existence and dimensions of a rotating white dwarf, J. Differential Equations 42 (1981), no. 3, 414 – 437. · Zbl 0493.76109 · doi:10.1016/0022-0396(81)90114-5
[17] H. Helmholtz, Über Integrate der hydrodynamischen Gleichungen, welche den Wirbetwegungen entsprechen, J. Reine Angew. Math. 55 (1858), 25-55. · ERAM 055.1448cj
[18] M. J. M. Hill, On a spherical vortex, Philos. Trans. Roy. Soc. London Ser. A 185 (1894), 213-245. · JFM 25.1471.01
[19] H. Lamb, Hydrodynamics, Cambridge, 1932. · JFM 26.0868.02
[20] Wei Ming Ni, On the existence of global vortex rings, J. Analyse Math. 37 (1980), 208 – 247. · Zbl 0457.76020 · doi:10.1007/BF02797686
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