Global solvability for the Boussinesq system with fractional Laplacian

This paper focuses on the global solvability for the Boussinesq system with fractional Laplacian $(-\mathrm{\Delta }{)}^{\alpha }$ in ${\mathbb{R}}^n$ for $n\ge 3$. It proves the existence of a small positive number $\epsilon =\epsilon (n,\alpha )$ such that for each $0<T<\mathrm{\infty }$, if $\frac{1}{2}<\alpha <\frac{2+n}{4}$ and $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{{\dot{H}}^{s_0}}+T^{1\mathrm{/}2}\mathrm{\parallel }{\theta }_0{\mathrm{\parallel }}_{{\dot{H}}^{s_0-\alpha }}\le \epsilon$, then the fractional Boussinesq system has a unique strong solution on the bounded interval $[0,T]$. If $\frac{1}{2}<\alpha <\frac{2+n}{6}$ and $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{{\dot{H}}^{s_0}}+\mathrm{\parallel }{\theta }_0{\mathrm{\parallel }}_{{\dot{H}}^{s_0-2\alpha }}\le \epsilon$, then the fractional Boussinesq system has a unique strong solution on the whole interval $[0,\mathrm{\infty })$.