Beale-Kato-Majda type criterion of smooth solutions for the three dimensional Boussinesq equations via partial horizontal derivatives of two velocity components

In this paper, we establish a blow-up criterion for the local smooth solutions to the three-dimensional incompressible Boussinesq equations in the framework of Besov spaces $ \dot{B}_{\infty ,\infty }^{0} $ via partial horizontal derivatives of two velocity components $ \partial _{1}u_{1} $ and $ \partial _{2}u_{2} $. Our result can be regarded as the borderline case of the work by F. Wu (Bull. Braz. Math. Soc., New Series (52) 2021, 267-279).