A regularity criterion of smooth solution for the 3D viscous Hall-MHD equations

In this work, we investigate the regularitycriterion for the solution of the Hall-MHD system in three-dimensions. It isproved that if the pressure $\pi $ and the gradient of magnetic field $%\nabla B$ satisfies some kind of space-time integrable condition on $[0,T]$,then the corresponding solution keeps smoothness up to time $T$. This resultimproves some previous works to the Morrey space $\overset{\cdot }{\mathcal{M}}_{2,\frac{3}{r}}$ for $0\leq r