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 Rⁿ for n≥3. It proves the existence of a small positive number ε=ε(n,α) such that for each 0<T<∞, if 1/2<α<2+n/4 and \|u₀\|_Ḣ^s₀+T1/2\|θ₀\|_Ḣ^s₀-α≤ ε, then the fractional Boussinesq system has a unique strong solution on the bounded interval $[0,T]$. If 1/2<α<2+n/6 and \|u₀\|_Ḣ^s₀+\|θ₀\|_Ḣ^s₀-2α≤ ε, then the fractional Boussinesq system has a unique strong solution on the whole interval [0,∞).