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On regularity criteria for the 3D magneto-micropolar fluid equations in the critical Morrey-Campanato space
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 |
$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] $.
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), 443-455.
doi: doi:10.1007/s002200050067. |
[2] |
Q. Chen, C. Miao and Z. Zhang, The Beale-Kato-Majda criterion for the 3D magneto-hydrodynamics equations, Comm. Math. Phys., 275 (2007), 861-872.
doi: doi:10.1007/s00220-007-0319-y. |
[3] |
G. Duvaut and J. L. Lions, Inéquations en thermoé lasticité et magnéto-hydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279. |
[4] |
S. Gala, A note on the uniqueness of mild solutions to the Navier-Stokes equations, Arch. Math.(Basel), 84 (2007), 448-454. |
[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), 105-108.
doi: doi:10.1016/0020-7225(77)90025-8. |
[6] |
C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.
doi: doi:10.1016/j.jde.2004.07.002. |
[7] |
T. Kato, Strong $L^p$ solutions of the Navier-Stokes equations in Morrey spaces, Bol. Soc. Bras. Mat.(N.S.), 22 (1992), 127-155. |
[8] |
P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem," Research Notes in Mathematics, Chapman & Hall, CRC, 2002. |
[9] |
P. G. Lemarié-Rieusset, The Navier-Stokes equations in the critical Morrey-Campanato space, Rev. Mat. Iberoam., 23 (2007), 897-930. |
[10] |
G. Lukaszewicz, "Micropolar Fluids: Theory and Applications," Birkhauser, Berlin 1998. |
[11] |
S. Machihara and T. Ozawa, Interpolation inequalities in Besov spaces, Proc. Amer. Math. Soc., 131 (2003), 1553-1556.
doi: doi:10.1090/S0002-9939-02-06715-1. |
[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), 75-90. |
[13] |
M. A. Rojas-Medar, Magneto-micropolar fluid motion: existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319.
doi: doi:10.1002/mana.19971880116. |
[14] |
M. A. Rojas-Medar and J. L. Boldrini, Magneto-micropolar fluid motion: existence of weak solutions, Rev. Mat. Complut., 11 (1998), 443-460. |
[15] |
M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.
doi: doi:10.1002/cpa.3160360506. |
[16] |
M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations, 17 (1992), 1407-1456.
doi: doi:10.1080/03605309208820892. |
[17] |
J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306.
doi: doi:10.1080/03605300701382530. |
[18] |
B. Yuan, Regularity of weak solutions to magneto-micropolar fluid equations, Acta Mathematica Scientia, 30 (2010), 1469-1480.
doi: doi:10.1016/S0252-9602(10)60139-7. |
[19] |
Y. Zhou, Remarks on regularities for the 3D MHD equations, Disc. Cont. Dyna. Sys., 12 (2005), 881-886.
doi: doi:10.3934/dcds.2005.12.881. |
[20] |
Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Internat. J. Non-Linear Mech., 41 (2006), 1174-1180.
doi: doi:10.1016/j.ijnonlinmec.2006.12.001. |
[21] |
Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505. |
[22] |
Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, To appear in Forum Math (2010), DOI 10.1515/FORM.2011.079. |
[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), 193-199.
doi: doi:10.1007/s00033-009-0023-1. |
[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), 3643-3648.
doi: doi:10.1016/j.na.2009.12.045. |
show all references
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), 443-455.
doi: doi:10.1007/s002200050067. |
[2] |
Q. Chen, C. Miao and Z. Zhang, The Beale-Kato-Majda criterion for the 3D magneto-hydrodynamics equations, Comm. Math. Phys., 275 (2007), 861-872.
doi: doi:10.1007/s00220-007-0319-y. |
[3] |
G. Duvaut and J. L. Lions, Inéquations en thermoé lasticité et magnéto-hydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279. |
[4] |
S. Gala, A note on the uniqueness of mild solutions to the Navier-Stokes equations, Arch. Math.(Basel), 84 (2007), 448-454. |
[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), 105-108.
doi: doi:10.1016/0020-7225(77)90025-8. |
[6] |
C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.
doi: doi:10.1016/j.jde.2004.07.002. |
[7] |
T. Kato, Strong $L^p$ solutions of the Navier-Stokes equations in Morrey spaces, Bol. Soc. Bras. Mat.(N.S.), 22 (1992), 127-155. |
[8] |
P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem," Research Notes in Mathematics, Chapman & Hall, CRC, 2002. |
[9] |
P. G. Lemarié-Rieusset, The Navier-Stokes equations in the critical Morrey-Campanato space, Rev. Mat. Iberoam., 23 (2007), 897-930. |
[10] |
G. Lukaszewicz, "Micropolar Fluids: Theory and Applications," Birkhauser, Berlin 1998. |
[11] |
S. Machihara and T. Ozawa, Interpolation inequalities in Besov spaces, Proc. Amer. Math. Soc., 131 (2003), 1553-1556.
doi: doi:10.1090/S0002-9939-02-06715-1. |
[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), 75-90. |
[13] |
M. A. Rojas-Medar, Magneto-micropolar fluid motion: existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319.
doi: doi:10.1002/mana.19971880116. |
[14] |
M. A. Rojas-Medar and J. L. Boldrini, Magneto-micropolar fluid motion: existence of weak solutions, Rev. Mat. Complut., 11 (1998), 443-460. |
[15] |
M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.
doi: doi:10.1002/cpa.3160360506. |
[16] |
M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations, 17 (1992), 1407-1456.
doi: doi:10.1080/03605309208820892. |
[17] |
J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306.
doi: doi:10.1080/03605300701382530. |
[18] |
B. Yuan, Regularity of weak solutions to magneto-micropolar fluid equations, Acta Mathematica Scientia, 30 (2010), 1469-1480.
doi: doi:10.1016/S0252-9602(10)60139-7. |
[19] |
Y. Zhou, Remarks on regularities for the 3D MHD equations, Disc. Cont. Dyna. Sys., 12 (2005), 881-886.
doi: doi:10.3934/dcds.2005.12.881. |
[20] |
Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Internat. J. Non-Linear Mech., 41 (2006), 1174-1180.
doi: doi:10.1016/j.ijnonlinmec.2006.12.001. |
[21] |
Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505. |
[22] |
Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, To appear in Forum Math (2010), DOI 10.1515/FORM.2011.079. |
[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), 193-199.
doi: doi:10.1007/s00033-009-0023-1. |
[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), 3643-3648.
doi: doi:10.1016/j.na.2009.12.045. |
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