Asymptotic stability of higher order norms in terms of asymptotic energy stability for viscous incompressible fluid flows heated from below

We consider a steady viscous incompressible fluid flow in an infinite layer heated from below. The steady flow is assumed to be periodic with respect to the plane variables. If this flow turns out to be asymptotically energy-stable with respect to a particular disturbance then it is also asymptotically stable in higher order norms with respect to the same perturbation. No smallness of the initial values is needed.