Well-posedness and convergence results for the 3D-Lagrange Boussinesq-α system

In this paper, we study the three-dimensional Lagrangian averaged Boussinesq- α system which is a regularized version of the three-dimensional Boussinesq system. We prove the existence of a weak solution to the 3D-Lagrangian averaged Boussinesq- α system, in Sobolev spaces. Unlike preceding works, this solution is global in time and depends continuously on the initial data, in particular, it is unique. More importantly, it converges to a weak solution of the three-dimensional Boussinesq system, as the regularizing parameter α vanishes.