Nonlinear energy stability in a compressible atmosphere

We provide sufficient conditions for nonlinear exponential stability of the compressible Benard problem. In particular, by using a generalized energy analysis we prove stability whenever the Rayleigh number does not exceed a computable critical number Rc. The value of Rc is given for finite amplitude depth and for thin layers as well, and such values are compared with those already computed in the linear theory. In the limit of depth which goes to zero a necessary and sufficient condition for nonlinear stability of the Benard problem is proved. The principle of exchange of stabilities is not required to hold