Blow-Up Criterion of the 3D Magnetic Bénard Equations via the Gradient of Pressure

In this paper, we address the problem raised by Q. Liu (Appl. Math. Lett. 104: 106255, 2020) to successfully establish a Serrin-type blow-up criterion for local smooth solutions to the 3D magnetic Bénard equations in terms of the gradient of pressure. More precisely, we prove that if the gradient of pressure \(\nabla P\) $\mathrm{\nabla }P$ satisfies \( \nabla P\in L^{\frac{2}{3-r}}(0,T; L^{\frac{3}{r}}(\mathbb {R}^{3}))\quad \text { with }~0<r\le 1, \) $$\mathrm{\nabla }P\in L^{\frac{2}{3-r}}(0,T;L^{\frac{3}{r}}({\mathbb{R}}^3))\text{with}0<r\le 1,$$ then the corresponding solution \((u,b,\theta )\) $(u,b,\theta )$ to the 3D magnetic Bénard equations can be extended beyond the time \(t= T\) $t=T$ .