Bifurcations of solutions for the Boussinesq system in T2T2

In this paper, we study the real analytic regularity and bifurcations of the solutions for the two-dimensional periodic Boussinesq system. If the critical points of the stream function ψψ are points of maxima or minima, one shall call these points viscous vortices because near these points the velocity uu is tangent to the level sets of ψψ which is a closed curve. For any smooth initial temperature θ0θ0, we can construct a set of initial stream functions such that initial critical points bifurcate from 1 to 2 and then to 3 critical points in finite time. The bifurcation takes place in a small neighborhood of the origin. For another set of initial stream functions, we show that 3 critical points merge into 1 critical point in finite time. We also construct a set of initial stream functions so that bifurcation can be generated without the existence of an initial critical point.