The BKM criterion to the 3D double-diffusive magneto convection systems involving planar components

In this paper, we investigate the BKM type blowup criterion applied to 3D double-diffusive magneto convection systems. Specifically, we demonstrate that a unique local strong solution does not experience blow-up at time $T$, given that $(\nabla_{h}\tilde{u}, \nabla_{h}\tilde{b})\in L^2(0,T;\dot{B}^{-1}_{\infty,\infty})$. To prove this, we employ the logarithmic Sobolev inequality in the Besov spaces with negative indices and a well-known commutator estimate established by Kato and Ponce. This result is the further improvement and extension of the previous works by O (2021) and Wu (2023).