Stability properties of the Boussinesq equations (Q1386190)

The stability of the equilibrium solutions of the dimensionless Boussinesq equation \[ \begin{aligned} \partial_tv & = \Delta v+ \sqrt R\theta k-\nabla \pi- (v\nabla)v \\ \text{Pr} \partial_t \vartheta & =\Delta \vartheta +\sqrt Rv_3- \text{Pr} (v\nabla) \vartheta,\;\text{div} v = 0 \end{aligned} \] with the boundary conditions \(v=0\), \(\vartheta =0\) on \(\partial \Omega\) is studied in an infinite slab. It is shown that there is a critical Rayleigh number \(R=R_C\) which is the boundary for stability in both the cases of periodic perturbations and \(L^2\) perturbations from the equilibrium solution. The results are shown to hold in the case of a roll (i.e. a perturbation which is periodic in one direction and constant in the other).