Global solvability for the Boussinesq system with fractional Laplacian
This paper focuses on the global solvability for the Boussinesq system with fractional Laplacian $(-Δ)^α$ in $\mathbb{R}^{n}$ for $n\geq3$. It proves the existence of a small positive number $\varepsilon=\varepsilon(n,α)$ such that for each $0<T<\infty$, if $\frac{1}{2}<α<\frac{2+n}{4}$ and $\|u_{0}\|_{\dot{H}^{s_{0}}}+T^{1/2}\|θ_{0}\|_{\dot{H}^{s_{0}-α}}\leq \varepsilon$, then the fractional Boussinesq system has a unique strong solution on the bounded interval $[0,T]$. If $\frac{1}{2}<α<\frac{2+n}{6}$ and $\|u_{0}\|_{\dot{H}^{s_{0}}}+\|θ_{0}\|_{\dot{H}^{s_{0}-2α}}\leq \varepsilon$, then the fractional Boussinesq system has a unique strong solution on the whole interval $[0,\infty)$.
math.AP↗