Global nonlinear stability in the Bénard problem for a mixture near the bifurcation point

The nonlinear stability of the motionless state of a binary fluid mixture heated and salted from below, in the Oberbeck-Boussinesq scheme, for stress-free and rigid-rigid boundary conditions and Schmidt numbers P_C greater than Prandtl numbers P_T, is studied in the region around the bifurcation point$\mathscr{C}_0^{2} = \mathscr{R}_B^{2}(P_T+1)/[P_T(p-1)]$ of linear instability. An improvement of the results in Mulone [11] is found for small values of p = P _C /P _T and P_T. For p sufficiently large the critical nonlinear Rayleigh number is very close to the linear one (with relative difference less than $1\%$ in the sea water case)