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

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