AIMS Mathematics, 2018, 3(4): 565-574. doi: 10.3934/Math.2018.4.565

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

1 ENS of Mostaganem, University of Mostaganem, Box 227, Mostaganem 27000, Algeria
2 Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria, 6 95125 Catania - Italy
3 RUDN University, 6 Miklukho - Maklay St, Moscow, 117198, Russia
4 Department of Mathematical Science , Faculty of Applied Science, Umm Alqura University, P. O.Box 14035, Makkah 21955, Saudi Arabia

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 $0\leq r<1$ which is larger than $L^{\frac{3}{r}}$.
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