The regularity criterion for weak solutions to the n-dimensional Boussinesq system

Abstract We consider the Boussinesq system in the homogeneous spaces of degree −1. To narrow the gap for the existence of small regular solutions in B ˙ ∞ , ∞ − 1 ( R n ) $\dot{B}^{-1}_{\infty,\infty}(\mathbb{R}^{n})$ , the biggest homogeneous space of degree −1 among those embedded in the space of tempered distributions, we show small solutions in the homogeneous Besov space B ˙ p , ∞ − 1 + n p ( R n ) $\dot{B}^{-1+\frac{n}{p}}_{p,\infty}(\mathbb{R}^{n})$ , with n ≥ 2 $n\geq2$ , n ≤ p < ∞ $n\leq p<\infty$