On the finite-dimensional global attractor for the Navier-Stokes-$\alpha$-Boussinesq model

Abstract

We consider a 3D regularized Boussinesq system with periodic boundary conditions, which is a particular case of the family of $\alpha$-models of turbulence, namely Navier-Stokes-$\alpha$-Boussinesq system. We first prove the global existence and uniqueness of weak solutions of the system. Then we give an upper bound for determining modes of weak solutions to the system. When the external force is time-independent, we prove the existence and Sobolev regularity of a finite-dimensional global attractor for the continuous semigroup associated to the system.