Global structure of 2-D incompressible flows

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 Poincaré-Bendixson to divergence-free vector fields on 2-manifolds of nonzero genus. Namely, the $\omega$ (or $\alpha$) 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 boundaries 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.