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On the motion of a vortex ring with a sharply concentrated vorticity. (English) Zbl 0956.35109
Authors’ abstract: We study an incompressible non-viscous fluid with axial symmetry without swirl, in the case when the vorticity is supported in an annulus. It is well known that there exists particular initial data for which the Euler evolution reduces to a translation with a constant speed. In this paper we prove a similar property for any initial condition in the limit situation in which the initial vorticity is sharply concentrated.

MSC:
35Q35 PDEs in connection with fluid mechanics
76B47 Vortex flows for incompressible inviscid fluids
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[1] Adebiyi, J. Mech. Appl. Math. XXXIV pp 153– (1981)
[2] Ambrosetti, Arch. Rat. Mech. Anal. 108 pp 97– (1989)
[3] Arms, Phys. Fluids 8 pp 553– (1965)
[4] An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, U.K., 1967. · Zbl 0152.44402
[5] Berger, Comm. Math. Phys. 77 pp 149– (1980)
[6] Fraenkel, Proc. Roy. Soc. Lond. A. 316 pp 29– (1970)
[7] Fraenkel, Acta Math. 132 pp 13– (1974)
[8] ?Variational Principles and Freeboundary Problems, Wiley, New York 1982.
[9] Fukumoto, J. Phys. Japan Soc. 55 pp 4152– (1986)
[10] Hama, Phys. Fluids 5 pp 1156– (1962)
[11] Hasimoto, J. Fluid Mech. 51 pp 477– (1972)
[12] Kida, J. Fluid Mech. 112 pp 397– (1981)
[13] Klein, Physica D 49 pp 323– (1991)
[14] Klein, Physica D 53 pp 267– (1991)
[15] Majda, Comm. Pure Appl. Math. 39 pp 187– (1986)
[16] Marchioro, Comm. Math. Phys. 116 pp 45– (1988)
[17] Marchioro, Math. Meth. in the Appl. Sci. 8 pp 328– (1986)
[18] Marchioro, Comm. Math. Phys 91 pp 563– (1983)
[19] and ?On the vortex-wave system?, in: Mechanics, Analysis and Geometry: 200 Years after Lagrange ( ed.), Elsevier Sciences, Amsterdam, 1991, pp. 79-95. · doi:10.1016/B978-0-444-88958-4.50007-0
[20] Marchioro, Comm. Math. Phys. 154 pp 49– (1993)
[21] and ?Mathematical theory of an incompressible nonviscous fluid?, Applied Mathematical Sciences 96, Springer, Heidelberg, 1994.
[22] Ni, J. Anal. Math. 37 pp 208– (1980)
[23] Nishiyama, Comm. Math. Phys. 162 pp 433– (1994)
[24] Shariff, Ann. Rev. Fluid Mech. 24 pp 235– (1992)
[25] Serfati, C.R. Acad. Sci., Paris 318 pp 925– (1994)
[26] Takaki, J. Phys. Japan Soc. 38 pp 1530– (1975)
[27] Turkington, Arch. Rat. Mech. Anal. 97 pp 75– (1987)
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