Linear and non-linear stability thresholds for thermal convection in a box

We analyse the problem of finding instability thresholds and global non-linear stability bounds for thermal convection in a linearly viscous fluid in a finite box. The vertical walls are maintained at different temperatures which gives rise to a non-uniform temperature field in steady state. This problem was previously analysed by Georgescu and Mansutti (Int. J. Non-Linear Mech. 1999; 34:603-613). In our work we determine the linear instability threshold to be well above the global stability boundary found by an energy method. Since the perturbed system is not symmetric we expect this to be the case, and our analysis yields a parameter region where possible subcritical instabilities may be found.