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AIMS Mathematics

2017, Volume 2, Issue 1: 16-23. doi: 10.3934/Math.2017.1.16
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Research article

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
  • Received: 21 October 2016 Accepted: 24 November 2016 Published: 09 December 2016

  • 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.

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

    Related Papers:

  • 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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