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We consider the Boussinesq system in the homogeneous spaces of degree −1. To narrow the gap for the existence of small regular solutions in \(\dot{B}^{-1}_{\infty,\infty}(\mathbb{R}^{n})\), 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 \(\dot{B}^{-1+\frac{n}{p}}_{p,\infty}(\mathbb{R}^{n})\), with \(n\geq2\), \(n\leq p<\infty\).

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