Interior structural bifurcation of 2D symmetric incompressible flows

Abstract

The structural bifurcation of a 2D divergence free vector field $ \mathbf{u}(\cdot, t) $ when $ \mathbf{u}(\cdot, t_0) $ has an interior isolated singular point $ \mathbf{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 $ \mathbf{u}(\cdot, t_0) $ is anti-symmetric with respect to $ \mathbf{x}_0 $, or symmetric with respect to the axis located on $ \mathbf{x}_0 $ and normal to the unique eigendirection of the Jacobian $ D\mathbf{u}(\cdot, t_0) $, the vector field must have index 1 or -1 at the singular point. Thus we study the structural bifurcation when $ \mathbf{u}(\cdot, t_0) $ has an interior isolated singular point $ \mathbf{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.