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

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  • 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.
    Mathematics Subject Classification: 35Q, 35B, 37L.

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