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

Hakima Bessaih, Benedetta Ferrario

Published 2015-04-20Version 1

In this paper, we study the 3D regularized Boussinesq equations. The velocity equation is regularized through a smoothing kernel of order $\alpha$ in the nonlinear term and with a $\beta$-fractional Laplacian; we are in the critical case $\alpha+\beta=\frac{5}{4}$. The temperature equation is a pure transport equation. We prove regularity results when the initial velocity is in $H^r$ and the initial temperature is in $H^{r-\beta}$ for $r>\max \left\{ \frac{5}{2}-2\alpha, \beta+1\right\}$ with $\beta\ge \frac{1}{2}$ and $\alpha\ge 0$. This regularity is enough to prove uniqueness of solutions. We also prove a continuous dependence of solutions with respect to the initial conditions.