Structure of bifurcated solutions of two-dimensional infinite Prandtl number convection with no-slip boundary conditions

摘要

We consider the two-dimensional infinite Prandtl number convection problem with no-slip boundary conditions. The existence of a bifurcation from the trivial solution to an attractor ΣR was proved by Park [13]. The no-stress case has been examined in [14]. We prove in this paper that the bifurcated attractor ΣR consists of only one cycle of steady state solutions and it is homeomorphic to S1. By thoroughly investigating the structure and transitions the solutions of the infinite Prandtl number convection problem in physical space, we confirm that the bifurcated solutions are indeed structurally stable. We show what the asymptotic structure of the bifurcated solutions looks like. This bifurcation analysis is based on a new notion of bifurcation, called attractor bifurcation, and structural stability is derived using geometric theories of incompressible flows.