Structure of 2D incompressible flows with the Dirichlet boundary conditions

We study in this article the structure and its stability of 2-D divergence-free vector fields with the Dirichlet boundary conditions. First we classify boundary points into two new categories: partial derivative-singular points and partial derivative-regular points, and establish an explicit formulation of divergence-free vector fields near the boundary. Second, local orbit structure near the boundary is classified. Then a structural stability theorem for divergence-free vector fields with the Dirichlet boundary conditions is obtained, providing necessary and sufficient conditions of a divergence-free vector fields. These structurally stability conditions are extremely easy to verify, and examples on stability of typical flow patterns are given. The main motivation of this article is to provide an important step for a forthcoming paper, where, for the first time, we are able to establish precise rigorous criteria on boundary layer separations of incompressible fluid flows, a long standing problem in fluid mechanics.