Structural classification and stability of divergence-free vector fields

It is proved that a divergence-free vector field v on a 2D submanifold of S2 is structurally stable with divergence-free vector field 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. The main motivation of this article and the companying articles [Indiana Univ. Math. J. 50 (1) (2001) 159; Appl. Math. Lett. 12 (1999) 39; Am. Math. Soc. 238 (1999) 193; Dynamics of 2D incompressible flows, in: P. Bates, K. Lu, D. Xu (Eds.), Differential Equations and Computational Simulations, World Scientific, Singapore, 2000, pp. 270–276; Math. Model. Numer. Anal. 34 (2) (2000) 419; Nonlin. Anal.: Real World Appl. 2 (2001) 467; Discrete Contin. Dyn. Syst. 7 (2) (2001) 431; Discrete Contin. Dyn. Syst., Ser. B 1 (1) (2001) 29] is to develop a geometric theory of 2D incompressible fluid flows in the physical spaces.