AIMS Mathematics, 2017, 2(1): 16-23. doi: 10.3934/Math.2017.1.16

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A logarithmically improved regularity criterion for the 3D MHD equations in Morrey-Campanato space

1 Department of Mathematics, University of Mostaganem, Box 227, Mostaganem, 27000, Algeria
2 Dipartimento di Matematicae Informatica, Universit`a di Catania Viale Andrea Doria, 6, 95125 Catania, Italy

In this paper, we will establish a sufficient condition for the regularity criterion to the 3D MHD equation in terms of the derivative of the pressure in one direction. It is shown that if the partial derivative of the pressure $\partial _{3}\pi $ satisfies the logarithmical Serrin type condition $\partial _{3}\pi $ satisfies the logarithmical Serrin type condition \begin{equation*} \int_{0}^{T}\frac{\left\Vert \partial _{3}\pi (s)\right\Vert _{\overset{% \cdot }{\mathcal{M}}_{2,\frac{3}{r}}}^{\frac{2}{2-r}}}{1+\ln (1+\left\Vert b(s)\right\Vert _{L^{4}})}ds<\infty \text{ for }0<1, \end{equation*} then the solution $(u,b)$ remains smooth on $\left[ 0,T\right] $. Compared to the Navier-Stokes result, there is a logarithmic correction involving $b$ in the denominator.
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Keywords MHD equations; regularity criteria

Citation: Sadek Gala, Maria Alessandra Ragusa. A logarithmically improved regularity criterion for the 3D MHD equations in Morrey-Campanato space. AIMS Mathematics, 2017, 2(1): 16-23. doi: 10.3934/Math.2017.1.16

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