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Dynamic bifurcation and stability in the Rayleigh-Bénard convection. (English) Zbl 1133.76315
Summary: We study in this article the bifurcation and stability of the solutions of the Boussinesq equations, and the onset of the Rayleigh-Benard convection. A nonlinear theory for this problem is established in this article using a new notion of bifurcation called attractor bifurcation and its corresponding theorem developed recently by the authors. This theory includes the following three aspects. First, 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, regardless of the multiplicity of the eigenvalue R c for the linear problem. Second, the bifurcated attractor A R is asymptotically stable. Third, when the spatial dimension is two, the bifurcated solutions are also structurally stable and are classified as well. In addition, the technical method developed provides a recipe, which can be used for many other problems related to bifurcation and pattern formation.

MSC:
76E06Convection (hydrodynamic stability)
37N10Dynamical systems in fluid mechanics, oceanography and meteorology