On the nonlinear stability of the thermodiffusive equilibrium for the magnetic Benard problem

In this paper we study the nonlinear Lyapunov stability of the thermodiffusive equilibrium of a viscous electroconducting horizontal uid layer heated from below. We reformulate the nonlinear stability problem, in terms of poloidal and toroidal fields, by projecting the initial perturbation evolution equations on some suitable orthogonal subspaces of the kinematically admissible functions. In such a way, if the principle of exchange of stabilities holds, we obtain, in the classical L^2-norm, the coincidence of linear and nonlinear stability bounds.