* Correspondence to: Prof. Dr. C. Marchioro, Dipartimento de Matematica, Universita` &&La Sapienza'', Piazzale Aido Moro 2, 1-00185 Roma, Italy. E-mail: Marchioro@axcasp.caspur.it Contract/grant sponsor: MURST Contract/grant sponsor: CNR-GNFM CCC 0170} 4214/2000/020147}22$17.50 Received 1 July 1998 Copyright  2000 John Wiley & Sons, Ltd. Mathematical Methods in the Applied Sciences Math. Meth. Appl. Sci., 23, 147}168 (2000) MOS subject classi"cation: 35 Q 05; 76 C 05 On the Motion of a Vortex Ring with a Sharply Concentrated Vorticity D. Benedetto, E. Caglioti and C. Marchioro* Dipartimento di Matematica, Universita ` **La Sapienza++, Piazzale A. Moro 2, 00185 Roma, Italy Communicated by H. Neunzert We study an incompressible non-viscous #uid with axial symmetry without swirl, in the case when the vorticity is supported in an annulus. It is well known that there exist 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. Copyright  2000 John Wiley & Sons, Ltd. 1. Introduction In this paper we discuss the time evolution of an incompressible non-viscous #uid moving in  with axial symmetry without swirl. We study the case in which the initial vorticity is supported in an annulus. This is a classical problem, widely studied since the last century and the object of present physical investigations (see for instance [24] and references quoted in). In particular, there are situations in which the motion leaves the shape of the vorticity invariant (the so-called &smoke rings'). Quite recently, the problem has been rigorously solved by variational methods [7] (see also [1, 2, 5, 6, 22] and references quoted in). The conclusion of these studies is that there exist particular initial data for which the Euler evolution reduces to a translation with a constant speed. Here, we prove a similar property for any initial condition in the limit situation in which the initial vorticity is sharply concentrated. The topic investigated in the present paper is a particular case of the problem of the time evolution of an incompressible non-viscous #uid moving in , when the initial vorticity is sharply concentrated around a line x (s, t) (where s denotes the arclength and t the time). In the case of planar symmetry the problem is already completely solved: x (s, t) reduces to a straight line, which remains at rest and the particles of #uid