Stability of higher norms in terms of energy-stability for the Boussinesq equations: Remarks on the asymptotic behaviour of convection-roll-type solutions (Q1328938)

The authors treat the asymptotic behaviour of convection-roll-type solutions in an infinite layer of a viscous incompressible fluid flow heated from below. The steady Boussinesq flow is assumed to be stable in the sense of Lyapunov with respect to the norm \(sup_{t \geq 0} E^{1/2} (t)\), where \(E(t)\) is the kinematic energy of the perturbation at time \(t\). The steady flow turns out to be stable in the sense of Lyapunov also with respect to \(L^ 2\)-norms including higher order derivatives. The authors indicate absorbing sets for disturbances with sufficiently small initial values. Some aspects of the time evolution of two-dimensional solutions for the Boussinesq equations are also considered. The paper does not present any numerical results and therefore appears to be only of academic interest.