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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

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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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# References

1. L. Berselli and G. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations, Proc. Amer. Math. Soc., 130 (2002), 3585-3595.

2. S. Benbernou, M Terbeche, and M.A. Ragusa, A logarithmically improved regularity criterion for the MHD equations in terms of one directional derivative of the pressure, Applicable Analysis, http://dx.doi.org/10.1080/00036811.2016.1207246.

3. C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274.

4. Q. Chen, C. Miao, and Z. Zhang, On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations, Comm. Math. Phys., 284 (2008), 919-930.

5. H. Duan, On regularity criteria in terms of pressure for the 3D viscous MHD equations, Appl.Anal., 91 (2012), 947-952.

6. G. Duvaut and J.-L. Lions, In´equations en thermo´elasticit´e et magn´etohydrodynamique, Arch.Ration. Mech. Anal., 46 (1972), 241-279.

7. S. Gala, Extension criterion on regularity for weak solutions to the 3D MHD equations, Math.Meth. Appl. Sci., 33 (2010), 1496-1503.

8. C. He and Y. Wang, Remark on the regularity for weak solutions to the magnetohydrodynamic equations, Math. Methods Appl. Sci., 31 (2008), 1667-1684.

9. L. Ni, Z. Guo, and Y. Zhou, Some new regularity criteria for the 3D MHD equations, J. Math.Anal. Appl., 396 (2012), 108-118.

10. C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J.Differential Equations, 213 (2005), 235-254.

11. E. Ji and J. Lee, Some regularity criteria for the 3D incompressible magnetohydrodynamics, J.Math. Anal. Appl., 369 (2010), 317-322.

12. X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations via partial derivatives, Kinet.Relat. Models, 5 (2012), 505-516.

13. X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations via partial derivatives, II. Kinet.Relat. Models, 7 (2014), no. 2, 291-304.

14. X. Jia and Y. Zhou, Ladyzhenskaya-Prodi-Serrin type regularity criteria for the 3D incompressible MHD equations in terms of 3 X 3 mixture matrices, Nonlinearity, 28 (2015), 3289-3307.

15. X. Jia and Y. Zhou, A new regularity criterion for the 3D incompressible MHD equations in terms of one component of the gradient of pressure, J. Math. Anal. Appl., 396 (2012), 345-350.

16. P.G. Lemari´e-Rieusset, The Navier-Stokes equations in the critical Morrey-Campanato space, Rev.Mat. Iberoam., 23 (2007), no. 3, 897-930.

17. H. Lin and L. Du, Regularity criteria for incompressible magnetohydrodynamics equations in three dimensions, Nonlinearity, 26 (2013), 219-239.

18. S. Machihara and T. Ozawa, Interpolation inequalities in Besov spaces, Proc. Amer. Math. Soc., 131 (2003), 1553-1556.

19. M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm.Pure Appl. Math., 36 (1983), 635-664.

20. J.Wu, Viscous and inviscid magnetohydrodynamics equations, J. Anal. Math., 73 (1997), 251-265.

21. J. Wu, Bounds and new approaches for the 3D MHD equations, J. Nonlinear Sci., 12 (2002), 395-413.

22. J.Wu, Regularity results for weak solutions of the 3D MHD equations, Discrete Contin. Dyn. Syst., 10 (2004), 543-556.

23. K. Yamazaki, Remarks on the regularity criteria of generalized MHD and Navier-Stokes systems, J. Math. Phys., 54 (2013), 011502, 16pp.

24. Z. Zhang, P. Li, and G. Yu, Regularity criteria for the 3D MHD equations via one directional derivative of the pressure, J. Math. Anal. Appl., 401 (2013), 66-71.

25. Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12(2005), 881-886.

26. Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Int. J. Non-Linear Mech., 41 (2006), 1174-1180.

27. Y. Zhou and S. Gala, Regularity criteria for the solutions to the 3D MHD equations in multiplier space, Z. Angrew. Math. Phys., 61 (2010), 193-199.

28. Y. Zhou and S. Gala, Regularity Criteria in Terms of the Pressure for the Navier-Stokes Equations in the Critical Morrey-Campanato Space, Z. Anal. Anwend., 30 (2011), 83-93.