A new approach to nonlinear L2-stability of double diffusive convection in porous media: Necessary and sufficient conditions for global stability via a linearization principle

A new approach to nonlinear L2-stability for double diffusive convection in porous media is given. An auxiliary system Σ of PDEs and two functionals V, W are introduced. Denoting by L and N the linear and nonlinear operators involved in Σ, it is shown that Σ-solutions are linearly linked to the dynamic perturbations, and that V and W depend directly on L-eigenvalues, while (along Σ) and not only depend directly on L-eigenvalues but also are independent of N. The nonlinear L2-stability (instability) of the rest state is reduced to the stability (instability) of the zero solution of a linear system of ODEs. Necessary and sufficient conditions for general, global L2-stability (i.e. absence of regions of subcritical instabilities for any Rayleigh number) are obtained, and these are extended to cover the presence of a uniform rotation about the vertical axis.