On the coupling between a Navier–Stokes equation and a total energy equation in spatial dimension N=3

We consider a class of nonlinear evolution systems, namely the Rayleigh–Benard equations. This system arises from the coupling between a Navier–Stokes equation for the velocity and the pressure and a total energy equation in spatial dimension N=3. We give a few existence results of solutions under suitable conditions in the right-hand side of the momentum equation, the forcing term depending on the temperature. To this end, we begin to solve an approximated problem, namely the Boussinesq system resulting from the Rayleigh–Benard equations through a fixed-point argument. Next, by a linear combination, we construct a new equivalent system. Finally, we give a priori estimates and compactness results before passing to the limit in the equivalent system.