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Attractor bifurcation theory and its applications to Rayleigh-Bénard convection. (English) Zbl 1210.37056
Summary: 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 ${𝒜}_{R}$ when the Rayleigh number $R$ crosses the first critical Rayleigh number ${R}_{c}$ for all physically sound boundary conditions.

##### MSC:
 37L30 Attractors and their dimensions, Lyapunov exponents 35B41 Attractors (PDE) 35Q35 PDEs in connection with fluid mechanics 37G35 Attractors and their bifurcations 37N10 Dynamical systems in fluid mechanics, oceanography and meteorology 76E06 Convection (hydrodynamic stability)