Global solvability for the Boussinesq system with fractional Laplacian

This paper focuses on the global solvability for the Boussinesq system with fractional Laplacian $(-\Delta)^{\alpha}$ in $\mathbb{R}^{n}$ for $n\geq3$. It proves the existence of a small positive number $\varepsilon=\varepsilon(n,\alpha)$ such that for each $0<T<\infty$, if $\frac{1}{2}<\alpha<\frac{2+n}{4}$ and $\|u_{0}\|_{\dot{H}^{s_{0}}}+T^{1/2}\|\theta_{0}\|_{\dot{H}^{s_{0}-\alpha}}\leq \varepsilon$, 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 $\|u_{0}\|_{\dot{H}^{s_{0}}}+\|\theta_{0}\|_{\dot{H}^{s_{0}-2\alpha}}\leq \varepsilon$, then the fractional Boussinesq system has a unique strong solution on the whole interval $[0,\infty)$.