American Institute of Mathematical Sciences

Advanced
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

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), 443-455. 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), 861-872. 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), 241-279.  Google Scholar

[4]

S. Gala, A note on the uniqueness of mild solutions to the Navier-Stokes equations, Arch. Math.(Basel), 84 (2007), 448-454.  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), 105-108. 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), 235-254. 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), 127-155.  Google Scholar

[8]

P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem," Research Notes in Mathematics, Chapman & Hall, CRC, 2002. Google Scholar

[9]

P. G. Lemarié-Rieusset, The Navier-Stokes equations in the critical Morrey-Campanato space, Rev. Mat. Iberoam., 23 (2007), 897-930.  Google Scholar

[10]

G. Lukaszewicz, "Micropolar Fluids: Theory and Applications," Birkhauser, Berlin 1998. Google Scholar

[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.  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), 75-90.  Google Scholar

[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.  Google Scholar

[14]

M. A. Rojas-Medar and J. L. Boldrini, Magneto-micropolar fluid motion: existence of weak solutions, Rev. Mat. Complut., 11 (1998), 443-460.  Google Scholar

[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.  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), 1407-1456. doi: doi:10.1080/03605309208820892.  Google Scholar

[17]

J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306. doi: doi:10.1080/03605300701382530.  Google Scholar

[18]

B. Yuan, Regularity of weak solutions to magneto-micropolar fluid equations, Acta Mathematica Scientia, 30 (2010), 1469-1480. 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), 881-886. 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), 1174-1180. 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. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.  Google Scholar

[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. 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), 193-199. 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), 3643-3648. doi: doi:10.1016/j.na.2009.12.045.  Google Scholar

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.  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), 861-872. 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), 241-279.  Google Scholar

[4]

S. Gala, A note on the uniqueness of mild solutions to the Navier-Stokes equations, Arch. Math.(Basel), 84 (2007), 448-454.  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), 105-108. 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), 235-254. 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), 127-155.  Google Scholar

[8]

P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem," Research Notes in Mathematics, Chapman & Hall, CRC, 2002. Google Scholar

[9]

P. G. Lemarié-Rieusset, The Navier-Stokes equations in the critical Morrey-Campanato space, Rev. Mat. Iberoam., 23 (2007), 897-930.  Google Scholar

[10]

G. Lukaszewicz, "Micropolar Fluids: Theory and Applications," Birkhauser, Berlin 1998. Google Scholar

[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.  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), 75-90.  Google Scholar

[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.  Google Scholar

[14]

M. A. Rojas-Medar and J. L. Boldrini, Magneto-micropolar fluid motion: existence of weak solutions, Rev. Mat. Complut., 11 (1998), 443-460.  Google Scholar

[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.  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), 1407-1456. doi: doi:10.1080/03605309208820892.  Google Scholar

[17]

J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306. doi: doi:10.1080/03605300701382530.  Google Scholar

[18]

B. Yuan, Regularity of weak solutions to magneto-micropolar fluid equations, Acta Mathematica Scientia, 30 (2010), 1469-1480. 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), 881-886. 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), 1174-1180. 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. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.  Google Scholar

[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. 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), 193-199. 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), 3643-3648. doi: doi:10.1016/j.na.2009.12.045.  Google Scholar

[1]

V. Afraimovich, J. Schmeling, Edgardo Ugalde, Jesús Urías. Spectra of dimensions for Poincaré recurrences. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 901-914. doi: 10.3934/dcds.2000.6.901
[2]

B. Fernandez, E. Ugalde, J. Urías. Spectrum of dimensions for Poincaré recurrences of Markov maps. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 835-849. doi: 10.3934/dcds.2002.8.835
[3]

Juan Wang, Xiaodan Zhang, Yun Zhao. Dimension estimates for arbitrary subsets of limit sets of a Markov construction and related multifractal analysis. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2315-2332. doi: 10.3934/dcds.2014.34.2315
[4]

Godofredo Iommi, Bartłomiej Skorulski. Multifractal analysis for the exponential family. Discrete & Continuous Dynamical Systems, 2006, 16 (4) : 857-869. doi: 10.3934/dcds.2006.16.857
[5]

V. Afraimovich, Jean-René Chazottes, Benoît Saussol. Pointwise dimensions for Poincaré recurrences associated with maps and special flows. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 263-280. doi: 10.3934/dcds.2003.9.263
[6]

Julien Barral, Yan-Hui Qu. On the higher-dimensional multifractal analysis. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 1977-1995. doi: 10.3934/dcds.2012.32.1977
[7]

Mario Roy, Mariusz Urbański. Multifractal analysis for conformal graph directed Markov systems. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 627-650. doi: 10.3934/dcds.2009.25.627
[8]

Zhihui Yuan. Multifractal analysis of random weak Gibbs measures. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5367-5405. doi: 10.3934/dcds.2017234
[9]

Luis Barreira. Dimension theory of flows: A survey. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345
[10]

Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469
[11]

Valentin Afraimovich, Jean-Rene Chazottes and Benoit Saussol. Local dimensions for Poincare recurrences. Electronic Research Announcements, 2000, 6: 64-74.

[12]

Zied Douzi, Bilel Selmi. On the mutual singularity of multifractal measures. Electronic Research Archive, 2020, 28 (1) : 423-432. doi: 10.3934/era.2020024
[13]

Mirela Domijan, Markus Kirkilionis. Graph theory and qualitative analysis of reaction networks. Networks & Heterogeneous Media, 2008, 3 (2) : 295-322. doi: 10.3934/nhm.2008.3.295
[14]

Balázs Bárány, Michaƚ Rams, Ruxi Shi. On the multifractal spectrum of weighted Birkhoff averages. Discrete & Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2021199
[15]

Jean-Pierre Francoise, Claude Piquet. Global recurrences of multi-time scaled systems. Conference Publications, 2011, 2011 (Special) : 430-436. doi: 10.3934/proc.2011.2011.430
[16]

Jerrold E. Marsden, Alexey Tret'yakov. Factor analysis of nonlinear mappings: p-regularity theory. Communications on Pure & Applied Analysis, 2003, 2 (4) : 425-445. doi: 10.3934/cpaa.2003.2.425
[17]

Lars Olsen. First return times: multifractal spectra and divergence points. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 635-656. doi: 10.3934/dcds.2004.10.635
[18]

Imen Bhouri, Houssem Tlili. On the multifractal formalism for Bernoulli products of invertible matrices. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1129-1145. doi: 10.3934/dcds.2009.24.1129
[19]

Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173
[20]

Eva Miranda, Romero Solha. A Poincaré lemma in geometric quantisation. Journal of Geometric Mechanics, 2013, 5 (4) : 473-491. doi: 10.3934/jgm.2013.5.473

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (81)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy