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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

Tian Ma 1, and  Shouhong Wang 2,

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

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