Structure of 2D incompressible flows with the Dirichlet boundary conditions
Tian Ma, Shouhong Wang · Discrete and Continuous Dynamical Systems - B · 2001
We study in this article the structure and its stability of 2-D divergence-freevector fields with the Dirichlet boundary conditions. First we classify boundarypoints into two new categories: $\partial$−singular points and $\partial$−regular points, and establishan explicit formulation of divergence-free vector fields near the boundary.Second, local orbit structure near the boundary is classified. Then a structural stabilitytheorem for divergence-free vector fields with the Dirichlet boundary conditionsis obtained, providing necessary and sufficient conditions of a divergence-free vectorfields. These structurally stability conditions are extremely easy to verify, and exampleson stability of typical flow patterns are given.The main motivation of this article is to provide an important step for a forthcomingpaper, where, for the first time, we are able to establish precise rigorous criteriaon boundary layer separations of incompressible fluid flows, a long standing problemin fluid mechanics.