On the energy functional for nonlinear stability of the classic Bénard problem

Nonlinear stability of motionless state of the classical Bénard problem in case of stress-free boundaries is studied for 2-dimensional disturbances, by the Liapunov’s second method. For Rayleigh number smaller than 27π^4 /4 the motionless state is proved to be unconditionally and exponentially stable with respect to a new Liapunov function which is essentially stronger than the kinetic energy