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.  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.  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.  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.  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.  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.  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.  doi: 10.1137/130937718.  Google Scholar

[8]

A. C. Eringen, Theory of micropolar fluids,, J. Math. Mech., 16 (1966), 1.   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.  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.  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.  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, (2002), 439.   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.  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.  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.  doi: 10.1088/0951-7715/19/2/012.  Google Scholar

[16]

G. Lukaszewicz, Micropolar Fluids, Theory and Applications,, Birkhäuser, (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.  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.  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.  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.  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.   Google Scholar

[22]

J. Wu, The generalized MHD equations,, J. Differential Equations, 195 (2003), 284.  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.  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.  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.   Google Scholar

[27]

K. Yamazaki, Global regularity of logarithmically supercritical MHD system with zero diffusivity,, Appl. Math. Lett., 29 (2014), 46.  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.  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).  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.  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.  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.  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.  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.  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.  doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar

show all references

References:
[1]

G. Ahmadi and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions,, Int. J. Engng. Sci., 12 (1974), 657.  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.  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.  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.  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.  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.  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.  doi: 10.1137/130937718.  Google Scholar

[8]

A. C. Eringen, Theory of micropolar fluids,, J. Math. Mech., 16 (1966), 1.   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.  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.  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.  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, (2002), 439.   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.  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.  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.  doi: 10.1088/0951-7715/19/2/012.  Google Scholar

[16]

G. Lukaszewicz, Micropolar Fluids, Theory and Applications,, Birkhäuser, (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.  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.  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.  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.  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.   Google Scholar

[22]

J. Wu, The generalized MHD equations,, J. Differential Equations, 195 (2003), 284.  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.  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.  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.   Google Scholar

[27]

K. Yamazaki, Global regularity of logarithmically supercritical MHD system with zero diffusivity,, Appl. Math. Lett., 29 (2014), 46.  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.  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).  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.  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.  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.  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.  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.  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.  doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar

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