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 à la Leray through a smoothing kernel of order α in the nonlinear term and a β -fractional Laplacian; we consider the critical case View the MathML source and we assume View the MathML source. The temperature equation is a pure transport equation, where the transport velocity is regularized through the same smoothing kernel of order α . We prove global well posedness when the initial velocity is in Hr and the initial temperature is in Hr−β for r>max⁡(2β,β+1). This regularity is enough to prove uniqueness of solutions. We also prove a continuous dependence of solutions on the initial conditions.