Optimality extension criteria to the 3D double-diffusive magneto convection system

Abstract

We prove the optimality extension criteria for the 3D double-diffusive
magneto convection system in terms of the planar components or its partial derivatives. More precisely, we show that a unique local strong solution does not blow up at time $T$ provided
$(\nabla_{h}\tilde{u}, \nabla_{h}\tilde{b})\in L^2(0,T;\dot{V}^{-1}_{\infty,\infty,2})$ or $(\tilde{u}, \tilde{b})\in L^2(0,T;\dot{V}^{0}_{\infty,\infty,2})$, where Vishik space $\dot{V}^{s}_{p,q,\theta}$
is larger than the homogeneous Besov space $\dot{B}^{s}_{p,q}$. Our method is based on a logarithmic interpolation inequality and the well-known Kato-Ponce commutator estimates.