Dynamic of rotating fluid layers: $L^2$-absorbing sets and onset of convection

Via the longtime behaviour of the perturbations to thermal conduction solution $m_0$, the nonlinear longtime behaviour of Navier-Stokes fluid mixtures filling horizontal rotating layers uniformly heated from below and salted by one salt -- either from above or below -- is investigated. Via the existence of $L^2-$absorbing sets, it is shown that the perturbations to $m_0$ are ultimately bounded. The onset of steady or oscillatory convection is analyzed. Via a Linearization Principle (Rionero, Rend. Lincei Mat. Appl., 2014) it is shown that the linear theory captures completely the physics of the problem since the linear stability implies the nonlinear global asymptotic stability in the $L^2-$norm.