CRITICAL RAYLEIGH NUMBER IN RAYLEIGH-BENARD CONVECTION

The Rayleigh-Benard convection is a classical problem in fluid dynamics. In the presence of rigid boundary condition, we identify the critical Rayleigh number R-a* by a reduced variational problem. We prove nonlinear asymptotic stability for motionless steady states for R-a < R-a*, and their nonlinear instability for R-a > R-a*. The dynamic of such instability is determined by the leading growing mode(s) for the corresponding linearized system within the time interval of instability.