Some new regularity criterion for MHD three-dimensional flow
Keywords:
3D MHD fluid, incompressible, porous medium, regularity criterion, weak solution.Abstract
The aim of the paper is to establish regularity criteria for the weak solution of fluid passing through the porous media in R^{3}. We show that if (grad_{h}u,\partial_{3}b_{3}) belongs to L^{2alpha,2gamma} with 2/alpha+3/gamma<=3, 1<=gamma<=infinity, then the weak solution is regular and unique; if (grad_{h}u, grad_{h}b) belongs to L^{2alpha,2gamma} with 2/alpha+3/gamma}<=3, 1<=gamma<=infinity, then the weak solution is regular and unique; if (partial_{3}u, grad u_{3}) belongs to L^{2alpha,2gamma} and (u_{3}, b, \partial_{3} b, grad b_{3}) belongsL^{4alpha,4gamma} with 2/alpha+3/gamma}<=3, 1<=gamma<=infinity, then the weak solution is regular and unique. Here we use the notation grad_{h}=(partial_{1}, partial_{2}).References
Beirao Da Veiga H. (1995). A new regularity class
for the Navier-Stokes equations in Rn. Chinese Annals
Mathematics, 16:407- 412.
Chae, D. & Choe, H. J. (1999). On the regularity criterion
for the solutions of the Navier-Stokes equations, Electron.
Journal of Differential Equations, 5:1 -7.
Chen, Q. Miao, C. & Zhang, Z. (2007). The Beale-
Kato-Majda criterion for the 3D magneto-hydrodynamics
equations, Communication Mathematical Physics,
:861 -872.
Darcy, H. (1856). Les fontaines publique de la ville de
Dijon. Paris: Dalmont.
Duan, H. (2012). On regularity Criteria in terms of
pressure for the 3D Viscous MHD equations, Applicable
Analysis, 91:947 -952.
Fan, J., Jiang, S., Nakamura, G. & Zhou, Y. (2011).
Logarithmically improved regularity criteria for the
Navier stokes and MHD equations, Janural Mathematical
Fluid Mechanics, 13:557 -571.
Forchheimer, P. (1901). Wasserbewegung durch Boden.
Zeitschrift des Vereines deutscher Ingenieure, 45:1736-
Gala, S., Sawano, Y. & Tanaka, H. (2012). On the
uniqueness of weak solutions of the 3D MHD equations
in the Orlic-Morrey space, Applicable Analysis, 18-,
iFirst.
He, C. & Wang, Y. (2008). Remark on the regularity for
weak solutions to the magnetohydr - dynamic equations,
Mathematical Methods in the Applied Sciences, 31:1667-
He, C. & Xin, Z. (2005). On the Regularity of Weak
Solution to the Magnetohydrodynamic Equations, Journal
of Differential Equations, 2:225- 254.
Ni, L. D., Guo, Z. G. & Zhou, Y. (2012). Some new
regularity criteria for the 3D MHD equations, Journal
Mathematical Analysis and Application, 396:108- 118.
Rahman, S. 2014. Regularity criterion for 3D MHD
fluid passing through the porous medium in terms of
gradient pressure, Journal of Computational and Applied
Mathematics , 270:88- 99.
Sermange, M. & Temam, R. (1983). Some mathematical
questions related to MHD equations, Communications on
Pure Applied Mathematics, 36:635 -664.
Zhou, Y. (2002). A new regularity criterion for the
Navier-Stokes equations in terms of the gradient of
velocity component, Methods and Applications Analysis,
- 578.
Zhou, Y. (2005). Remarks on regularities for the 3D
MHD equations, Discrete Continuous Dynamics System,
:881- 886.
Zhou, Y. (2006). Regularity criteria for the 3D MHD
equations in terms of pressure, Non-Linear Mechanics,
:1174 -1180.
Zhou Y. & Fan, J. (2012). Logarithmically improved
regularity criteria for the 3D viscous MHD equations,
Forum Mathematics, 24:691 -708.
Beg, I., Saha, M., Ganguly, A. & Dey, D. (2014).
Random fixedpoint of Gregus mapping and its application
to nonlinear stochastic integral equations, kuwait Journal
of Science, 41:1- 14.
