The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion

In this paper, we study the 3D regularized Boussinesq equations. The velocity equation is regularized a la Leray through a smoothing kernel of order alpha in the nonlinear term and a beta-fractional Laplacian; we consider the critical case alpha + beta =5/4 and we assume 1/2< beta <5/4. The temperature equation is a pure transport equation, where the transport velocity is regularized through the same smoothing kernel of order alpha. We prove global well posedness when the initial velocity is in H-r and the initial temperature is in Hr-beta for r > max(2 beta, beta + 1). This regularity is enough to prove uniqueness of solutions. We also prove a continuous dependence of solutions on the initial conditions. (C) 2016 Elsevier Inc. All rights reserved.