Almost exponential decay of Benard convection problem without surface tension

We consider the dynamics of an Boussinesq approximation Benard convection uid evolving in a three-dimensional domain bounded below by a xed atten boundary and above by a free moving surface. The domain is horizontally periodic and the eect of the surface tension is neglected on the free surface. By developing a priori estimates for the model, we prove the global existence and almost exponential decay of solutions in the framework of high regularity.