On the principle of exchange of stabilities in Rayleigh--Bénard convection (Q2706083)

The problem of Rayleigh-Bénard convection is considered in a general situation when the heat source and gravity field are variable. For the stress-free boundary conditions it is proved that the principle of exchange of stabilities holds as long as the product of gravity field and the integral of the heat sources is nonnegative throughout the layer. The proof is based on the theory of positive operators and uses the positivity properties of Green's function.