We investigate stability properties of the following three steady flows in an infinite fluid layer: 1. The motionless state of the Rayleigh-Bénard convection; 2. plane Couette flow in a rotating layer; and 3. plane Couette flow in a rotating layer heated from below. These steady flows are proved to be unconditionally asymptotically stable under 2-D periodic perturbations even when the control parameters reach their critical values for the linearized stability. The proof is carried out based on the Ljapunov function method.
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