Sharp global nonlinear stability for temperature-dependent viscosity convection

With regard to the dependence of viscosity on temperature, an unconditional non-linear energy-stability analysis for thermal convection according to Navier-Stokes theory has not yet been developed. We here analyse three models of fluid behaviour proposed by O. A. Ladyzhenskaya. We show that by using these theories we can develop an unconditional analysis directly. Two of the models lead directly to an unconditional development employing L2 theory, while the third necessitates the introduction of a generalized energy also involving an L3 term. The nonlinear stability boundaries in all three cases are sharp when compared with the instability thresholds of linear theory.