DISCRETE AND CONTINUOUS Website: http://AIMsciences.org DYNAMICAL SYSTEMS Volume 7, Number 2, April 2001 pp. 431–445 GLOBAL STRUCTURE OF 2-D INCOMPRESSIBLE FLOWS Tian Ma Department of Mathematics Sichuan University Chengdu, P. R. China Shouhong Wang Department of Mathematics Indiana university Bloomington, IN 47405 Dedicated to Professor Chen Wenyuan on the Occasion of his 70th Birthday Abstract. The main objective of this article is to classify the structure of divergence- free vector fields on general two-dimensional compact manifold with or without boundaries. First we prove a Limit Set Theorem, Theorem 2.1, a generalized version of the Poincar´ e-Bendixson to divergence-free vector fields on 2-manifolds of nonzero genus. Namely, the ω (or α) limit set of a regular point of a regular divergence-free vector field is either a saddle point, or a closed orbit, or a closed domain with bound- aries consisting of saddle connections. We call the closed domain ergodic set. Then the ergodic set is fully characterized in Theorem 4.1 and Theorem 5.1. Finally, we obtain a global structural classification theorem (Theorem 3.1), which amounts to saying that the phase structure of a regular divergence-free vector field consists of finite union of circle cells, circle bands, ergodic sets and saddle connections. Introduction. The main motivation of this article long with [1, 4, 5, 6, 7, 8] is to develop a geometric theory of 2-D incompressible fluid flows. Our main philosophy is to classify the topological structure and its transitions of the instantaneous ve- locity field (i.e. streamlines in the Eulerian coordinates), treating the time variable as a parameter. Based on this philosophy, our project contains studies in two di- rections: 1) the development of a global geometric theory of divergence-free vector fields on general two-dimensional compact manifolds with or without boundary, and 2) the connections between the solutions of the Navier-Stokes (or Euler) equations and the dynamics of the velocity fields in the physical space. This article along with [4, 7, 8] is in the first direction. In [4, 7], we studied the case where M is a compact sub-manifold of S 2 . In this case, the classical Poincar´ e- Bendixson theorem holds true. Hence in particular, we proved that a divergence-fee vector field is structurally stable with divergence-free vector fields perturbations if and only if (1) v is regular; (2) all interior saddle points of v are self-connected; and (3) each boundary saddle point is connected to boundary saddles on the same connected component of the boundary. These conditions are quite different from those in the classical M. Peixoto [11] structural stability theorem for general vector 1991 Mathematics Subject Classification. 34D, 35Q35, 58F, 76, 86A10. The work was supported in part by the Office of Naval Research under Grant N00014-96-1- 0425, by the National Science Foundation under Grants DMS-9623071 and DMS-0072612, and by the National Science Foundation of China under Grant 19971062. 431