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 in [6]. This
theory includes the following three aspects. First, the problem bifurcates from the trivial solution
an attractor AR when the Rayleigh number R crosses the first critical Rayleigh number
Rc for all physically sound boundary conditions, regardless of the multiplicity of the eigenvalue
Rc for the linear problem.
Second, the bifurcated attractor AR 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.