On regularity criteria for the 3D magneto-micropolar fluid equations in the critical Morrey-Campanato space

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March 2011, 10(2): 583-592. doi: 10.3934/cpaa.2011.10.583

On regularity criteria for the 3D magneto-micropolar fluid equations in the critical Morrey-Campanato space

Jinbo Geng ^1, , Xiaochun Chen ^2, and Sadek Gala ^3,

1.	Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China
2.	College of Mathematics and Computer Science, Chongqing Three Gorges University, Wanzhou 404000, Chongqing, China
3.	Department of Mathematics, University of Mostaganem, Box 227, Mostaganem 27000, Algeria

Received May 2010 Revised September 2010 Published December 2010

Abstract
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In this paper, some improved regularity criteria for the 3D magneto-micropolar fluid equations are established in critical Morrey-Campanato spaces. It is proved that if the velocity field satisfies

$u\in L^{\frac{2}{1-r}}(0,T; M_{2,\frac{3}{r}}(R^3)) $ with $r\in (0, 1)$ or $u\in C(0, T; M_{2,3}(R^3))$

or the gradient field of velocity satisfies

$ \nabla u\in L^{\frac{2}{2-r}}(0, T; M_{2,\frac{3}{ r}}(R^3))$ with $r\in (0,1], $

then the solution remains smooth on $[0,T] $.

Keywords: multifractal analysis., Poincaré recurrences, Dimension theory.

Mathematics Subject Classification: Primary: 35Q35, 35B65; Secondary: 76D0.

Citation: Jinbo Geng, Xiaochun Chen, Sadek Gala. On regularity criteria for the 3D magneto-micropolar fluid equations in the critical Morrey-Campanato space. Communications on Pure & Applied Analysis, 2011, 10 (2) : 583-592. doi: 10.3934/cpaa.2011.10.583

References:

[1]	R. E. Caflisch, I. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD,, Comm. Math. Phys., 184 (1997), 443. doi: doi:10.1007/s002200050067. Google Scholar
[2]	Q. Chen, C. Miao and Z. Zhang, The Beale-Kato-Majda criterion for the 3D magneto-hydrodynamics equations,, Comm. Math. Phys., 275 (2007), 861. doi: doi:10.1007/s00220-007-0319-y. Google Scholar
[3]	G. Duvaut and J. L. Lions, Inéquations en thermoé lasticité et magnéto-hydrodynamique,, Arch. Rational Mech. Anal., 46 (1972), 241. Google Scholar
[4]	S. Gala, A note on the uniqueness of mild solutions to the Navier-Stokes equations,, Arch. Math.(Basel), 84 (2007), 448. Google Scholar
[5]	G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluids equations,, Internat. J. Engrg. Sci., 15 (1977), 105. doi: doi:10.1016/0020-7225(77)90025-8. Google Scholar
[6]	C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations,, J. Differential Equations, 213 (2005), 235. doi: doi:10.1016/j.jde.2004.07.002. Google Scholar
[7]	T. Kato, Strong $L^p$ solutions of the Navier-Stokes equations in Morrey spaces,, Bol. Soc. Bras. Mat.(N.S.), 22 (1992), 127. Google Scholar
[8]	P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem,", Research Notes in Mathematics, (2002). Google Scholar
[9]	P. G. Lemarié-Rieusset, The Navier-Stokes equations in the critical Morrey-Campanato space,, Rev. Mat. Iberoam., 23 (2007), 897. Google Scholar
[10]	G. Lukaszewicz, "Micropolar Fluids: Theory and Applications,", Birkhauser, (1998). Google Scholar
[11]	S. Machihara and T. Ozawa, Interpolation inequalities in Besov spaces,, Proc. Amer. Math. Soc., 131 (2003), 1553. doi: doi:10.1090/S0002-9939-02-06715-1. Google Scholar
[12]	E. Ortega-Torres and M. A. Rojas-Medar, On the uniqueness and regularity of the weak solutions for magneto-micropolar equations,, Rev. Mat. Apl., 17 (1996), 75. Google Scholar
[13]	M. A. Rojas-Medar, Magneto-micropolar fluid motion: existence and uniqueness of strong solution,, Math. Nachr., 188 (1997), 301. doi: doi:10.1002/mana.19971880116. Google Scholar
[14]	M. A. Rojas-Medar and J. L. Boldrini, Magneto-micropolar fluid motion: existence of weak solutions,, Rev. Mat. Complut., 11 (1998), 443. Google Scholar
[15]	M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, Comm. Pure Appl. Math., 36 (1983), 635. doi: doi:10.1002/cpa.3160360506. Google Scholar
[16]	M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations,, Comm. Partial Differential Equations, 17 (1992), 1407. doi: doi:10.1080/03605309208820892. Google Scholar
[17]	J. Wu, Regularity criteria for the generalized MHD equations,, Comm. Partial Differential Equations, 33 (2008), 285. doi: doi:10.1080/03605300701382530. Google Scholar
[18]	B. Yuan, Regularity of weak solutions to magneto-micropolar fluid equations,, Acta Mathematica Scientia, 30 (2010), 1469. doi: doi:10.1016/S0252-9602(10)60139-7. Google Scholar
[19]	Y. Zhou, Remarks on regularities for the 3D MHD equations,, Disc. Cont. Dyna. Sys., 12 (2005), 881. doi: doi:10.3934/dcds.2005.12.881. Google Scholar
[20]	Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure,, Internat. J. Non-Linear Mech., 41 (2006), 1174. doi: doi:10.1016/j.ijnonlinmec.2006.12.001. Google Scholar
[21]	Y. Zhou, Regularity criteria for the generalized viscous MHD equations,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 24 (2007), 491. Google Scholar
[22]	Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations,, To appear in Forum Math (2010), (2010). Google Scholar
[23]	Y. Zhou and S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space,, Z. Angew. Math. Phys., 61 (2010), 193. doi: doi:10.1007/s00033-009-0023-1. Google Scholar
[24]	Y. Zhou and S. Gala, A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field. Nonlinear Anal,, Nonlinear Anal., 72 (2010), 3643. doi: doi:10.1016/j.na.2009.12.045. Google Scholar

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References:

[1]	R. E. Caflisch, I. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD,, Comm. Math. Phys., 184 (1997), 443. doi: doi:10.1007/s002200050067. Google Scholar
[2]	Q. Chen, C. Miao and Z. Zhang, The Beale-Kato-Majda criterion for the 3D magneto-hydrodynamics equations,, Comm. Math. Phys., 275 (2007), 861. doi: doi:10.1007/s00220-007-0319-y. Google Scholar
[3]	G. Duvaut and J. L. Lions, Inéquations en thermoé lasticité et magnéto-hydrodynamique,, Arch. Rational Mech. Anal., 46 (1972), 241. Google Scholar
[4]	S. Gala, A note on the uniqueness of mild solutions to the Navier-Stokes equations,, Arch. Math.(Basel), 84 (2007), 448. Google Scholar
[5]	G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluids equations,, Internat. J. Engrg. Sci., 15 (1977), 105. doi: doi:10.1016/0020-7225(77)90025-8. Google Scholar
[6]	C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations,, J. Differential Equations, 213 (2005), 235. doi: doi:10.1016/j.jde.2004.07.002. Google Scholar
[7]	T. Kato, Strong $L^p$ solutions of the Navier-Stokes equations in Morrey spaces,, Bol. Soc. Bras. Mat.(N.S.), 22 (1992), 127. Google Scholar
[8]	P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem,", Research Notes in Mathematics, (2002). Google Scholar
[9]	P. G. Lemarié-Rieusset, The Navier-Stokes equations in the critical Morrey-Campanato space,, Rev. Mat. Iberoam., 23 (2007), 897. Google Scholar
[10]	G. Lukaszewicz, "Micropolar Fluids: Theory and Applications,", Birkhauser, (1998). Google Scholar
[11]	S. Machihara and T. Ozawa, Interpolation inequalities in Besov spaces,, Proc. Amer. Math. Soc., 131 (2003), 1553. doi: doi:10.1090/S0002-9939-02-06715-1. Google Scholar
[12]	E. Ortega-Torres and M. A. Rojas-Medar, On the uniqueness and regularity of the weak solutions for magneto-micropolar equations,, Rev. Mat. Apl., 17 (1996), 75. Google Scholar
[13]	M. A. Rojas-Medar, Magneto-micropolar fluid motion: existence and uniqueness of strong solution,, Math. Nachr., 188 (1997), 301. doi: doi:10.1002/mana.19971880116. Google Scholar
[14]	M. A. Rojas-Medar and J. L. Boldrini, Magneto-micropolar fluid motion: existence of weak solutions,, Rev. Mat. Complut., 11 (1998), 443. Google Scholar
[15]	M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, Comm. Pure Appl. Math., 36 (1983), 635. doi: doi:10.1002/cpa.3160360506. Google Scholar
[16]	M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations,, Comm. Partial Differential Equations, 17 (1992), 1407. doi: doi:10.1080/03605309208820892. Google Scholar
[17]	J. Wu, Regularity criteria for the generalized MHD equations,, Comm. Partial Differential Equations, 33 (2008), 285. doi: doi:10.1080/03605300701382530. Google Scholar
[18]	B. Yuan, Regularity of weak solutions to magneto-micropolar fluid equations,, Acta Mathematica Scientia, 30 (2010), 1469. doi: doi:10.1016/S0252-9602(10)60139-7. Google Scholar
[19]	Y. Zhou, Remarks on regularities for the 3D MHD equations,, Disc. Cont. Dyna. Sys., 12 (2005), 881. doi: doi:10.3934/dcds.2005.12.881. Google Scholar
[20]	Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure,, Internat. J. Non-Linear Mech., 41 (2006), 1174. doi: doi:10.1016/j.ijnonlinmec.2006.12.001. Google Scholar
[21]	Y. Zhou, Regularity criteria for the generalized viscous MHD equations,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 24 (2007), 491. Google Scholar
[22]	Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations,, To appear in Forum Math (2010), (2010). Google Scholar
[23]	Y. Zhou and S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space,, Z. Angew. Math. Phys., 61 (2010), 193. doi: doi:10.1007/s00033-009-0023-1. Google Scholar
[24]	Y. Zhou and S. Gala, A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field. Nonlinear Anal,, Nonlinear Anal., 72 (2010), 3643. doi: doi:10.1016/j.na.2009.12.045. Google Scholar

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