# American Institute of Mathematical Sciences

May  2006, 6(3): 591-604. doi: 10.3934/dcdsb.2006.6.591

## Dynamic bifurcation theory of Rayleigh-Bénard convection with infinite Prandtl number

 1 Department of Mathematics & Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, United States

Received  March 2005 Revised  October 2005 Published  February 2006

We study in this paper the bifurcation and stability of the solutions of the Rayleigh-Bénard convection which has the infinite Prandtl number, using a notion of bifurcation called attractor bifurcation. We prove that the problem bifurcates from the trivial solution to an attractor $\A_R$ when the Rayleigh number $R$ crosses the critical Rayleigh number $R_c$. As a special case, we also prove another result which corresponds to the classical pitchfork bifurcation, that this bifurcated attractor $\A_R$ consists of only two stable steady states when the first eigenvalue $R_1$ is simple.
Citation: Jungho Park. Dynamic bifurcation theory of Rayleigh-Bénard convection with infinite Prandtl number. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 591-604. doi: 10.3934/dcdsb.2006.6.591
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