EVENT DETAILS AND ABSTRACT

Title: Dynamics of point vortices in the plane and on the sphere: the non- linear stability of relative equilibria
Speaker: Stefanella Boatto , Institute of Celestial Mechanics and Institute Henri Poincare
Speaker Info: Paris
Brief Description:
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Abstract:

In 1883, while studying the atomic structure, J.J. Thomson investigated the linear stability of corotating point vortices in the plane. His interest was in configurations of identical vortices equally spaced along the circumference of a circle, i.e. located at the vertices of a regular polygon. He proved that for six or fewer vortices the polygonal configurations are stable, while for seven vortices - the Thomson heptagon - he erroneously concluded that the configuration is slightly unstable. It took more than a century to make some progress on this problem! In 1985, Dritschel proved that the Thomson heptagon is neutrally stable and that for eight or more vortices the corresponding polygonal configurations are linearly unstable. Recently (1999) Cabral & Schmidt proved that for seven or fewer vortices the polygonal configurations are non-linearly stable in the plane. For the spherical case, results are much more recent! In 1993 Dritschel & Polvani determined the ranges of linear stability - in terms of the latitude- of polygonal configurations. By a method similar to that used by Dritschel in the planar case, Dritschel & Polvani showed that at the pole, for N<7 the configuration is stable, for N=7 it is neutrally stable and for N>7 it is unstable. In 1998 Marsden & Pekarsky proved that for N=3 the range of non linear stability is the whole sphere (including results for vortices with different vorticities k₁, k₂ and k₃). Cabral and I (2001) determined the ranges of non-linear stability for all N.