Structural evolution of the Taylor vortices

We classify in this article the structure and its transitions/évolution of the Taylor vortices with perturbations in one of the following catégories: a) the Hamiltonian vector fields, b) the divergencefree vector fields, and c). the solutions of the Navier-Stokes équations on the two-dimensional torus. This is part of a project orient ed toward to developing a geometrie theory of incompressible fluid flows in the physical spaces. Résumé. Dans cet article, nous classons la structure et les transit ions/évolutions des vort ex de Taylor avec perturbations dans l'une des catégories suivantes : a) champs de vecteurs hamiltoniens, b) champs de vecteurs à divergence nulle, et c) solutions des équations de Navier-Stokes sur le tore bidimensionnel. Cette partie du projet est orientée vers une théorie géométrique des écoulements de fluides incompressibles dans Vespace physique. Mathematics Subject Classification. 34D, 35Q35, 58F, 76. Received: September 17, 1999. Revised: December 14, 1999. INTRODUCTION The Taylor vector fields, or simply the Taylor fields, are referred to the divergence-tree vector fields vnrn on the two-dimensional torus M = T = E/(2TTZ) defined by vn7Yl = (jn cosnx\ cosmx2,n sinnrri sinma^), (0-1) where n, m > 1 are integers. The Taylor vortices are referred to the periodic structures of the phase diagram of the Taylor fields illustrated by Figure 2.2. It is easy to see that the Taylor fields are Hamiltonian vector fields whose Hamiltonian functions (or the stream functions) are given by Hnrn(xi,X2) = cosnxi. sinmx2. By the Hodge décomposition, Hamiltonian vector fields on the torus as well as in a gênerai 2-manifold do not exhaust all divergence-free vector fields. The study of the Taylor vortices is originated in Taylor's 1923 [49]. In f act, such periodic structure appears in many problems of mathematics and physics. The Taylor vortex type of periodic structures appear also in the solutions of many partial difTerential équations; see Bensoussan, Lions and Papanicolaou [4]. We would like to