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 $\alpha+\beta=\frac{5}{4}$ and we assume $\frac 12 <\beta<\frac 54$. 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 $H^{r-\beta}$ for $r>\max(2\beta,\beta+1)$. This regularity is enough to prove uniqueness of solutions. We also prove a continuous dependence of the solutions on the initial conditions.