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 Ḃ∞ ,∞⁰ via partial horizontal derivatives of two velocity components ∂ ₁u₁ and ∂ ₂u₂. 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).