Stability of idgher norms in terms of energy-stability for the Boussinesq-equations: Remarks on the asymptotic behaviour of convection-roll-type solutions

In an infinite layer heated from below we perturb a steady viscous incomprssible fluid flow. The steady flow is assumed to be stable in the sense of Lyapunov with respect to the norm sup_t≥0 E^1/2 (t), where E(t) is the kinetic energy of the perturbation at timet. Energy-stability may also set in at later times. Then the steady flow turns out to be stable in the sense ofLyapunov also with respect to L²-norms of higher order derivatives. In terms of these we are able to indicate absorbing sets for the disturbances if their initial values are small. The smallness required is discussed. As a by-product we obtain that (nonlinear) instability of any steady flow with respect to higher order norms of the disturbances, as considered in [1] by Galdi and Padula for perturbations of the motionless state in various situations, implies (nonlinear) instability with respect to kinetic energy, at least. Finally we consider some aspects of the time evolution of two-dimensional solutions of the Boussinesq-equations. Two-dimensional means that the velocity-field and the temperature do not depend on one of the plane space-variables. We speak of convection-roll-type solutions.