Bifurcation of Critical Points for Solutions of the 2D Euler and 2D Quasi-geostrophic Equations

Abstract

We consider the 2D Euler and 2D quasi-geostrophic equations with periodic boundary conditions. For both systems we will use the stream-function formulation and study the bifurcation problem for the critical points of the stream function. In a small neighborhood of the origin, we construct a set of initial data such that initial critical points of the stream function bifurcate from 1 to 2 and then to 3 critical points in finite time. For the quasi-geostrophic equation the whole bifurcation process takes place strictly within the lifespan of the constructed local solution. © 2012 Springer Science+Business Media, LLC.