Asymptotic study to strong solution of a 3d regularization to boussinesq system in sobolev spaces

A regularized periodic three-dimensional Boussinesq system is studied. The existence, uniqueness and continuous dependance with respect to the initial data of weak and strong solutions are proved under the minimum regularity requirements. The main novelty is that these solutions are global in time. Also, convergence results of the unique weak solution and the unique strong solution of the three-dimensional regularized Boussinesq system to solutions of the three-dimensional Boussinesq system are established as the regularizing parameter a vanishes. We overcome the main difficulty caused by the singular dependance of the energy estimates on the regularizing parameter; as if it vanishes, the energy estimates blow up. The proofs use energy methods and compactness arguments.