On space-time derivative estimates for the magnetohydrodynamic equations

In this paper, we present the space-time derivative estimates of solutions to the MHD equations. It is a generalization of a priori bounds for the classical Navier-Stokes system by Duff in \cite[Acta Math. 164, 1990]{[Duff]} and gives an affirmative answer to a question proposed by Zheligovsky in \cite[Mathematics. 9, 2021]{[Zheligovsky]}. In addition, we show that $u,B\in L^{\text{f}qq-3}(0,T;L^q({\mathbb{R}}^3))$ with $6\le q\le \mathrm{\infty }$ and $D^{k}u,D^{k}B\in L^{\text{f}qq(k+1)-3}(0,T;L^q({\mathbb{R}}^3))$ for $k\ge 1,2\le q\le \mathrm{\infty }$ in this system.