Global Existence and Decaying Rates of the Strong Solution for the Boussinesq System

This paper focuses on the global existence and time-decay rates of the strong solution for the Boussinesq system with full viscosity in ${\mathbb{R}}^n$ for $n\ge 3$ . Under the initial assumption of $\left({\theta }_0,u_0\right)\in L^{n/3}\times L^n$ with a small norm, and $n>3$ or $n=3$ and ${\theta }_0\in L^{r_0}$ for some $r_0>1$ , global existence and uniqueness of the strong solution $\left(\theta ,u\right)$ for the Boussinesq system is established. This solution is proven to obey the following estimates: ${‖\theta \left(t\right)‖}_r\le C{t}^{-\left(3-n/p\right)/2}$ for $n/3\le p<\infty$ , ${‖u\left(t\right)‖}_p\le C{t}^{-\left(1-n/q\right)/2}$ for $n\le q\le \infty$ , ${‖\nabla \theta \left(t\right)‖}_p\le C{t}^{-\left(3-n/p\right)/2-1/2}$ and ${‖{\nabla }^2\theta \left(t\right)‖}_p=O\left(t^{-n\left(1/r-1/p\right)/2-1}\right)$ as $t⟶\infty$ for $r\le p<n/2$ , and ${‖\nabla u\left(t\right)‖}_q\le C{t}^{-\left(1-n/q\right)/2-1/2}$ and ${‖{\nabla }^{2}u\left(t\right)‖}_q=O\left(t^{-n\left(1/r-1/q\right)/2-1}\right)$ as $t⟶\infty$ for $n\le q<2n$ , where $r=n/3$ if $n>3$ and $1<r<\min \left\{r_0,n/2\right\}$ if $n=3$ .