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.
Keywords: Divergence-free vector fields, structural stability, Dirichlet boundary
Revised: January 2001; Published: January 2001.
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