Research article

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

  • 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.


    加载中
    [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, Inéquations en thermoélasticité et magnétohydrodynamique, 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. Lemarié-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.
  • Reader Comments
  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3754) PDF downloads(990) Cited by(2)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog