Abstract
We study the Cauchy problem of certain Boussinesq-alpha equations in n dimensions with n = 2 or 3. We establish regularity for the solution under del u is an element of L(1)(0, T; (B)over dot(infinity,infinity)(0) (R(n))) As a corollary, the smooth solution of the Leray-alpha-Boussinesq system exists globally, when n = 2. For the Lagrangian averaged Boussinesq equations, a regularity criterion del theta is an element of L(1)(0, T; L(infinity)(R(2))) is established. Other Boussinesq systems with partial viscosity are also discussed in the paper.