Global solvability for the Boussinesq system with fractional Laplacian

This paper focuses on the global solvability for the Boussinesq system with fractional Laplacian (-) ^ in R^n for n3. It proves the existence of a small positive number = (n, ) such that for each 0<T<, if 12<<2+n4 and \|u₀\|₇̇^ₒ_{₀}+T^1/2\|₀\|₇̇^ₒ_{₀-}, then the fractional Boussinesq system has a unique strong solution on the bounded interval 0, T. If 12<<2+n6 and \|u₀\|₇̇^ₒ_{₀}+\|₀\|₇̇^ₒ_{₀-2}, then the fractional Boussinesq system has a unique strong solution on the whole interval [0, ).