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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: $\partial$−singular points and $\partial$−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.