Global well-posedness and optimal decay rates for the $n$-D incompressible Boussinesq equations with fractional dissipation and thermal diffusion

In this paper, $n$-dimensional incompressible Boussinesq equations with fractional dissipation and thermal diffusion are investigated. Firstly, by applying frequency decomposition, we find that $\Vert (u,\theta ){\Vert }_{L^2({\mathbb{R}}^n)}\to 0$, as $t\to \mathrm{\infty }$. Secondly, by using energy methods, we can show that if the initial data is sufficiently small in $H^s({\mathbb{R}}^n)$ with s = 1+ $\frac{n}{2}-2\alpha (0<\alpha <1)$, the global solutions are derived. Furthermore, under the assumption that the initial data $(u_0$, ${\theta }_0)$ belongs to $L^p($where $1\le p<2)$, using a more advanced frequency decomposition method, we establish optimal decay estimates for the solutions and their higher-order derivatives. Meanwhile, the uniqueness of the system can be obtained. In the case $\alpha$ = 0, we obtained the regularity and decay estimate of the damped Boussinesq equation in Besov space.