# American Institute of Mathematical Sciences

December  2014, 7(4): 779-792. doi: 10.3934/krm.2014.7.779

## $(N-1)$ velocity components condition for the generalized MHD system in $N-$dimension

 1 Department of Mathematics, Washington State University, Pullman, WA 99164-3113, United States

Received  March 2014 Revised  August 2014 Published  November 2014

We study the magnetohydrodynamics system, generalized via a fractional Laplacian. When the domain is in $N-$dimension, $N$ being three, four or five, we show that the regularity criteria of its solution pair may be reduced to $(N-1)$ many velocity field components with the improved integrability condition in comparison to the result in [29]. Furthermore, we extend this result to the three-dimensional magneto-micropolar fluid system.
Citation: Kazuo Yamazaki. $(N-1)$ velocity components condition for the generalized MHD system in $N-$dimension. Kinetic & Related Models, 2014, 7 (4) : 779-792. doi: 10.3934/krm.2014.7.779
##### References:
 [1] G. Ahmadi and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions, Int. J. Engng. Sci., 12 (1974), 657-663. doi: 10.1016/0020-7225(74)90042-1.  Google Scholar [2] J. Beale, T. Kato and A. Majda, Remarks on breakdown of smooth solutions for the three-dimensional Euler equations, Comm. Math. Phys., 94 (1984), 61-66. doi: 10.1007/BF01212349.  Google Scholar [3] S. Benbernou, S. Gala and M. A. Ragusa, On the regularity criteria for the 3D magnetohydrodynamic equations via two components in terms of BMO space, Math. Methods Appl. Sci., 37 (2013), 2320-2325. doi: 10.1002/mma.2981.  Google Scholar [4] L. C. 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. doi: 10.1090/S0002-9939-02-06697-2.  Google Scholar [5] C. Cao and E. S. Titi, Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57 (2008), 2643-2661. doi: 10.1512/iumj.2008.57.3719.  Google Scholar [6] C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274. doi: 10.1016/j.jde.2009.09.020.  Google Scholar [7] C. Cao, J. Wu and B. Yuan, The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J. Math. Anal., 46 (2014), 588-602. doi: 10.1137/130937718.  Google Scholar [8] A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  Google Scholar [9] S. Gala, Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey-Campanato space, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 181-194. doi: 10.1007/s00030-009-0047-4.  Google Scholar [10] G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Int. J. Engng. Sci., 15 (1977), 105-108. doi: 10.1016/0020-7225(77)90025-8.  Google Scholar [11] C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254. doi: 10.1016/j.jde.2004.07.002.  Google Scholar [12] H. Inoue, K. Matsuura and M. Ŏtani, Strong solutions of magneto-micropolar fluid equation, in Discrete and continuous dynamical systems, 2003; Dynamical systems and differential equations, Wilmington, NC (2002), 439-448.  Google Scholar [13] X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations via partial derivatives, II, Kinet. Relat. Models, 7 (2014), 291-304. doi: 10.3934/krm.2014.7.291.  Google Scholar [14] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.  Google Scholar [15] I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 453-469. doi: 10.1088/0951-7715/19/2/012.  Google Scholar [16] G. Lukaszewicz, Micropolar Fluids, Theory and Applications, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar [17] E. E. Ortega-Torres and M. A. Rojas-Medar, Magneto-micropolar fluid motion: Global existence of strong solutions, Abstr. Appl. Anal., 4 (1999), 109-125. doi: 10.1155/S1085337599000287.  Google Scholar [18] P. Penel and M. Pokorný, On anisotropic regularity criteria for the solutions to 3D Navier-Stokes equations, J. Math. Fluid Mech., 13 (2011), 341-353. doi: 10.1007/s00021-010-0038-6.  Google Scholar [19] M. A. Rojas-Medar, Magneto-micropolar fluid motion: Existence and uniqueness of strong solutions, Math. Nachr., 188 (1997), 301-319. doi: 10.1002/mana.19971880116.  Google Scholar [20] M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.  Google Scholar [21] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187-195.  Google Scholar [22] J. Wu, The generalized MHD equations, J. Differential Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.  Google Scholar [23] J. Wu, Global regularity for a class of generalized magnetohydrodynamic equations, J. Math. Fluid Mech., 13 (2011), 295-305. doi: 10.1007/s00021-009-0017-y.  Google Scholar [24] N. Yamaguchi, Existence of global strong solution to the micropolar fluid system in a bounded domain, Math. Meth. Appl. Sci., 28 (2005), 1507-1526. doi: 10.1002/mma.617.  Google Scholar [25] K. Yamazaki, Regularity criteria of porous media equation in terms of one partial derivative or pressure field,, Commun. Math. Sci., ().   Google Scholar [26] K. Yamazaki, Regularity criteria of supercritical beta-generalized quasi-geostrophic equation in terms of partial derivatives, Electron. J. Differential Equations, 2013 (2013), 1-12.  Google Scholar [27] K. Yamazaki, Global regularity of logarithmically supercritical MHD system with zero diffusivity, Appl. Math. Lett., 29 (2014), 46-51. doi: 10.1016/j.aml.2013.10.014.  Google Scholar [28] K. Yamazaki, Regularity criteria of MHD system involving one velocity component and one current density component, J. Math. Fluid Mech., 16 (2014), 551-570. doi: 10.1007/s00021-014-0178-1.  Google Scholar [29] K. Yamazaki, Remarks on the regularity criteria of three-dimensional magnetohydrodynamics system in terms of two velocity field components, J. Math. Phys., 55 (2014), 031505, 16pp. doi: 10.1063/1.4868277.  Google Scholar [30] B. Yuan, Regularity of weak solutions to magneto-micropolar fluid equations, Acta Math. Sci. Ser. B Engl. Ed., 30 (2010), 1469-1480. doi: 10.1016/S0252-9602(10)60139-7.  Google Scholar [31] B. Yuan, On regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space, Proc. Amer. Math. Soc., 138 (2010), 2025-2036. doi: 10.1090/S0002-9939-10-10232-9.  Google Scholar [32] J. Yuan, Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations, Math. Meth. Appl. Sci., 31 (2008), 1113-1130. doi: 10.1002/mma.967.  Google Scholar [33] Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12 (2005), 881-886. doi: 10.3934/dcds.2005.12.881.  Google Scholar [34] Y. Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in $\mathbbR^{3}$, Proc. Amer. Math. Soc., 134 (2006), 149-156. doi: 10.1090/S0002-9939-05-08312-7.  Google Scholar [35] Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. I. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505. doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar

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##### References:
 [1] G. Ahmadi and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions, Int. J. Engng. Sci., 12 (1974), 657-663. doi: 10.1016/0020-7225(74)90042-1.  Google Scholar [2] J. Beale, T. Kato and A. Majda, Remarks on breakdown of smooth solutions for the three-dimensional Euler equations, Comm. Math. Phys., 94 (1984), 61-66. doi: 10.1007/BF01212349.  Google Scholar [3] S. Benbernou, S. Gala and M. A. Ragusa, On the regularity criteria for the 3D magnetohydrodynamic equations via two components in terms of BMO space, Math. Methods Appl. Sci., 37 (2013), 2320-2325. doi: 10.1002/mma.2981.  Google Scholar [4] L. C. 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. doi: 10.1090/S0002-9939-02-06697-2.  Google Scholar [5] C. Cao and E. S. Titi, Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57 (2008), 2643-2661. doi: 10.1512/iumj.2008.57.3719.  Google Scholar [6] C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274. doi: 10.1016/j.jde.2009.09.020.  Google Scholar [7] C. Cao, J. Wu and B. Yuan, The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J. Math. Anal., 46 (2014), 588-602. doi: 10.1137/130937718.  Google Scholar [8] A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  Google Scholar [9] S. Gala, Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey-Campanato space, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 181-194. doi: 10.1007/s00030-009-0047-4.  Google Scholar [10] G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Int. J. Engng. Sci., 15 (1977), 105-108. doi: 10.1016/0020-7225(77)90025-8.  Google Scholar [11] C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254. doi: 10.1016/j.jde.2004.07.002.  Google Scholar [12] H. Inoue, K. Matsuura and M. Ŏtani, Strong solutions of magneto-micropolar fluid equation, in Discrete and continuous dynamical systems, 2003; Dynamical systems and differential equations, Wilmington, NC (2002), 439-448.  Google Scholar [13] X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations via partial derivatives, II, Kinet. Relat. Models, 7 (2014), 291-304. doi: 10.3934/krm.2014.7.291.  Google Scholar [14] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.  Google Scholar [15] I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 453-469. doi: 10.1088/0951-7715/19/2/012.  Google Scholar [16] G. Lukaszewicz, Micropolar Fluids, Theory and Applications, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar [17] E. E. Ortega-Torres and M. A. Rojas-Medar, Magneto-micropolar fluid motion: Global existence of strong solutions, Abstr. Appl. Anal., 4 (1999), 109-125. doi: 10.1155/S1085337599000287.  Google Scholar [18] P. Penel and M. Pokorný, On anisotropic regularity criteria for the solutions to 3D Navier-Stokes equations, J. Math. Fluid Mech., 13 (2011), 341-353. doi: 10.1007/s00021-010-0038-6.  Google Scholar [19] M. A. Rojas-Medar, Magneto-micropolar fluid motion: Existence and uniqueness of strong solutions, Math. Nachr., 188 (1997), 301-319. doi: 10.1002/mana.19971880116.  Google Scholar [20] M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506.  Google Scholar [21] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187-195.  Google Scholar [22] J. Wu, The generalized MHD equations, J. Differential Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.  Google Scholar [23] J. Wu, Global regularity for a class of generalized magnetohydrodynamic equations, J. Math. Fluid Mech., 13 (2011), 295-305. doi: 10.1007/s00021-009-0017-y.  Google Scholar [24] N. Yamaguchi, Existence of global strong solution to the micropolar fluid system in a bounded domain, Math. Meth. Appl. Sci., 28 (2005), 1507-1526. doi: 10.1002/mma.617.  Google Scholar [25] K. Yamazaki, Regularity criteria of porous media equation in terms of one partial derivative or pressure field,, Commun. Math. Sci., ().   Google Scholar [26] K. Yamazaki, Regularity criteria of supercritical beta-generalized quasi-geostrophic equation in terms of partial derivatives, Electron. J. Differential Equations, 2013 (2013), 1-12.  Google Scholar [27] K. Yamazaki, Global regularity of logarithmically supercritical MHD system with zero diffusivity, Appl. Math. Lett., 29 (2014), 46-51. doi: 10.1016/j.aml.2013.10.014.  Google Scholar [28] K. Yamazaki, Regularity criteria of MHD system involving one velocity component and one current density component, J. Math. Fluid Mech., 16 (2014), 551-570. doi: 10.1007/s00021-014-0178-1.  Google Scholar [29] K. Yamazaki, Remarks on the regularity criteria of three-dimensional magnetohydrodynamics system in terms of two velocity field components, J. Math. Phys., 55 (2014), 031505, 16pp. doi: 10.1063/1.4868277.  Google Scholar [30] B. Yuan, Regularity of weak solutions to magneto-micropolar fluid equations, Acta Math. Sci. Ser. B Engl. Ed., 30 (2010), 1469-1480. doi: 10.1016/S0252-9602(10)60139-7.  Google Scholar [31] B. Yuan, On regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space, Proc. Amer. Math. Soc., 138 (2010), 2025-2036. doi: 10.1090/S0002-9939-10-10232-9.  Google Scholar [32] J. Yuan, Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations, Math. Meth. Appl. Sci., 31 (2008), 1113-1130. doi: 10.1002/mma.967.  Google Scholar [33] Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12 (2005), 881-886. doi: 10.3934/dcds.2005.12.881.  Google Scholar [34] Y. Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in $\mathbbR^{3}$, Proc. Amer. Math. Soc., 134 (2006), 149-156. doi: 10.1090/S0002-9939-05-08312-7.  Google Scholar [35] Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. I. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505. doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar
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