Abstract
Analytical study of the regularization of the Boussinesq system is performed in frequency space using Fourier theory. Existence and uniqueness of weak solutions with minimum regularity requirement are proved. Convergence results of the unique weak solution of the regularized Boussinesq system to a weak Leray-Hopf solution of the Boussinesq system are established as the regularizing parameter a vanishes. The proofs are done in the frequency space and use energy methods, the Arzela-Ascoli compactness theorem and a Friedrichs-like approximation scheme.