Attractor bifurcation theory and its applications to Rayleigh-Bénard convection

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December 2003, 2(4): 591-599. doi: 10.3934/cpaa.2003.2.591

Attractor bifurcation theory and its applications to Rayleigh-Bénard convection

1.	Department of Mathematics, Sichuan University, Chengdu, China
2.	Department of Mathematics, Indiana University, Bloomington, IN 47405, United States

Revised October 2003 Published October 2003

Abstract
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In this note, we present a fast communication of a new bifurcation theory for nonlinear evolution equations, and its application to Rayleigh-Bénard Convection. The proofs of the main theorems presented will appear elsewhere. The bifurcation theory is based on a new notion of bifurcation, called attractor bifurcation. We show that as the parameter crosses certain critical value, the system bifurcates from a trivial steady state solution to an attractor with dimension between $m$ and $m+1$, where $m+1$ is the number of eigenvalues crosses the imaginary axis. Based on this new bifurcation theory, we obtain a nonlinear theory for bifurcation and stability of the solutions of the Boussinesq equations, and the onset of the Rayleigh-Bénard convection. In particular, we show that the problem bifurcates from the trivial solution an attractor $\mathcal A_R$ when the Rayleigh number $R$ crosses the first critical Rayleigh number $R_c$ for all physically sound boundary conditions.

Keywords: asymptotic stability, central manifold theory, Attractor bifurcation, semigroup, roll structure., structural stability, Rayleigh-Bénard convection.

Mathematics Subject Classification: 37L, 37G, 35Q, 58F, 76E, 76.

Citation: Tian Ma, Shouhong Wang. Attractor bifurcation theory and its applications to Rayleigh-Bénard convection. Communications on Pure & Applied Analysis, 2003, 2 (4) : 591-599. doi: 10.3934/cpaa.2003.2.591

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