On the Lyapunov function for some convection problems in hydrodynamic

In this paper we study the nonlinear Lyapunov stability of the mechanical equilibrium for a fluid mixture in a plane layer, in presence of linear skewsymmetric effects, such as the Coriolis term in the rotating Bénard problem. If we reformulate the nonlinear stability problem, projecting the initial perturbation evolution equations on some suitable orthogonal subspaces, we preserve the contribution of the skewsymmetric term, and jointly all the nonlinear terms vanish, indipendently from the boundary conditions. In this way we recover some sufficient conditions of nonlinear stability that are necessary and sufficient conditions of linear stability too. Because of this we can not require from the beginning that some functionals occurring in the energy relation must be definite positive. After studying the problem we can select only the results where such functionals satisfy the previous requisites, by using the linear instability analysis.