Linear stability of equilibria of a fluid that is a nonconductor of heat

Convective stability is studied in the linear approximation of equilibria of a strongly viscous fluid that is a nonconductor of heat where the fluid fills a bounded domain in a gravitational field. The corresponding system consists of the heat equation with transport in a velocity field and the steady-state Stokes system for the velocity and pressure. The latter includes an Archimedean force proportional to the temperature.
It is proved that equilibria for which the temperature strictly increases upward are stable in $L_2$ with respect to the temperature and in $W_2^2$ with respect to the velocity. Here, however, perturbations may die out arbitrarily slowly (Banach–Steinhaus stability). Under rough violation of the condition of monotonicity of the temperature the equilibrium is unstable.
Some critical cases of stability are also considered.