INTERIOR STRUCTURAL BIFURCATION OF 2D SYMMETRIC INCOMPRESSIBLE FLOWS

Abstract

The structural bifurcation of a 2D divergence free vector field u(., t) when u(., t(0)) has an interior isolated singular point x(0) of zero index has been studied by Ma and Wang [23]. Although in the class of divergence free fields which undergo a local bifurcation around a singular point, the ones with index zero singular points are generic, this class excludes some important families of symmetric flows. In particular, when u(., t(0)) is anti-symmetric with respect to x(0), or symmetric with respect to the axis located on x(0) and normal to the unique eigendirection of the Jacobian Du(., t(0)), the vector field must have index 1 or -1 at the singular point. Thus we study the structural bifurcation when u(., t(0)) has an interior isolated singular point x(0) with index -1, 1. In particular, we show that if such a vector field with its acceleration at t(0) both satisfy the aforementioned symmetries then generically the flow will undergo a local bifurcation. Under these generic conditions, we rigorously prove the existence of flow patterns such as pairs of co-rotating vortices and double saddle connections. We also present numerical evidence of the Stokes flow in a rectangular cavity showing that the bifurcation scenarios we present are indeed realizable.