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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:
37L30Attractors and their dimensions, Lyapunov exponents
35B41Attractors (PDE)
35Q35PDEs in connection with fluid mechanics
37G35Attractors and their bifurcations
37N10Dynamical systems in fluid mechanics, oceanography and meteorology
76E06Convection (hydrodynamic stability)