ABSTRACT
The magnetic Bénard problem concerns the onset of convection in a horizontal layer of a heavy, incompressible, viscous, electrically conducting fluid, heated from below, when a vertical constant magnetic field is imposed and the Boussinesq approximation is adopted. In this case, the rest state is a possible solution for all values of the temperature gradient β and of the intensity of the magnetic field H; however, such basic solution is observable only for suitable combinations of the values β and H. While he linear stability studies the limit relation between such values, that is, the critical linear Rayleigh number as function of the Hartman number Q L , above which the convection sets in, the nonlinear stability looks for a corresponding limit relation between critical nonlinear Rayleigh number and the Hartman number Q n , below which the rest is nonlinearly stable. The nonlinear stability of magnetic Bénard problem has been studied with different methods, and is based on the choice of an appropriate Lyapunov functional. The original energy method furnishes for the same critical number as for a non electrically conducting fluid. We shall call ”classical energy” the Lyapunov functional constituted only by the L 2 norms of the variables. The first heuristic reasoning that gives a as an increasing function of Q n is due to Galdi 1985 [2]. Then, a more general method of construction has been proposed by Galdi and Padula 1990 [3], and finally Rionero and Mulone [5] have considered a Lyapunov function that improves the earlier nonlinear stability results. Such methods, also called “generalized energy methods”, require the use of a heavy norm in the energy functional containing the derivatives of the basic variables, also called “generalized energy”. Till the paper [2], the only results which let the temperature gradient to increase with the increasing of the basic magnetic field where those due to Chandrasekhar for the linearized system.
