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December  2016, 9(6): 2167-2179. doi: 10.3934/dcdss.2016090

Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space

Baoquan Yuan 1, and  Xiao Li 2,

1. 

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, Henan
2. 

School of Mathematics and Information Science, Henan Polytechnic University, Henan 454000, China

Received  July 2015 Revised  September 2016 Published  November 2016

In this paper, we investigate the blow-up criteria of smooth solutions and the regularity of weak solutions to the micropolar fluid equations in three dimensions. We obtain that if $ \nabla_{h}u,\nabla_{h}\omega\in L^{1}(0,T;\dot{B}^{0}_{\infty,\infty})$ or $ \nabla_{h}u,\nabla_{h}\omega\in L^{\frac{8}{3}}(0,T;\dot{B}^{-1}_{\infty,\infty})$ then the solution $(u,\omega)$ can be extended smoothly beyond $t=T$.
Keywords: Micropolar fluid equations, blow-up criteria, regularity criteria, smooth solutions, Besov space..
Mathematics Subject Classification: Primary: 35Q35, 35B65; Secondary: 76N1.
Citation: Baoquan Yuan, Xiao Li. Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2167-2179. doi: 10.3934/dcdss.2016090
References:
[1]

L. C. Berselli, On a regularity criterion for the solutions to 3D Navier-Stokes equations, Differential Integral Equations, 15 (2002), 1129-1137.  Google Scholar

[2]

H. Bahouri, R. Danchin and J. Y. Chemin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer Heidelberg Dordrecht London New York, Springer-Verlag Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

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Q. L. Chen and C. X. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differential Equations, 252 (2012), 2698-2724. doi: 10.1016/j.jde.2011.09.035.  Google Scholar

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B. Q. Dong and Z. M. Chen, Regularity criteria of weak solutions to the three-dimensional micropolar flows, J. Math. Phys., 50 (2009), 103525, 13 pp. doi: 10.1063/1.3245862.  Google Scholar

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B. Q. Dong and W. Zhang, On the regularity criterion for the three-dimensional micropolar flows in Besov spaces, Nonlinear Anal., 73 (2010), 2334-2341. doi: 10.1016/j.na.2010.06.029.  Google Scholar

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B. Q. Dong and Z. F. Zhang, The BKM criterion for the 3D Navier-Stokes equations via two velocity components, Nonlinear Anal., 11 (2010), 2415-2421. doi: 10.1016/j.nonrwa.2009.07.013.  Google Scholar

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A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  Google Scholar

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D. Y. Fang and C. Y. Qian, Regularity criteria for 3D Navier-Stokes equations in Besov space, Comm.Pura Appl, 13 (2014), 585-603. doi: 10.3934/cpaa.2014.13.585.  Google Scholar

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S. Gala, On regularity criteria for the three-dimensional micropolar fluid equations in the critical Morrey-Campanato space, Nonlinear Anal., 12 (2011), 2142-2150. doi: 10.1016/j.nonrwa.2010.12.028.  Google Scholar

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I. Kukavica and M. Zinae, Navier-Stokes equation with regularity in one direction, J. Math. Phys., 48 (2007), 065203, 10 pp. doi: 10.1063/1.2395919.  Google Scholar

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P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hill/CRC, 2002. doi: 10.1201/9781420035674.  Google Scholar

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P. L. Lions and Lions, Mathematical Topics in Fluid Mechanics, Oxford University Press Inc. New York, 1996.  Google Scholar

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G. Lukaszewicz, Micropolar Fluids, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

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G. Lukaszewicz, On nonstationary flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat. (5), 12 (1988), 83-97.  Google Scholar

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E. Ortega-Torres and M. Rojas-Medar, On the regularity for solutions of the micropolar fluid equations, Rend. Semin. Mat. Univ. Padova, 122 (2009), 27-37. doi: 10.4171/RSMUP/122-3.  Google Scholar

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M. A. Rojas-Medar and J. L. Boldrini, Magneto-micropolar fluid motion: Existence of weak solutions, Rev. Mat. Complut., 11 (1998), 443-460. doi: 10.5209/rev_REMA.1998.v11.n2.17276.  Google Scholar

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Z. Skalák, Criteria for the regularity of the solutions to the Navier-Stokes equations based on the velocity gradient, Nonlinear Anal., 118 (2015), 1-21. doi: 10.1016/j.na.2015.01.011.  Google Scholar

[23]

Y. X. Wang and H. J. Zhao, Logarithmically improved blow up criterion for smooths solution to the 3D Micropolar Fluid equations, J. Appl. Math., (2012), Art. ID 541203, 13 pp. doi: 10.1016/j.nonrwa.2011.12.018.  Google Scholar

[24]

N. Yamaguchi, Existence of global strong solution to the micropolar fluid system in a bounded domain, Methods Appl. Sci., 28 (2005), 1507-1526. doi: 10.1002/mma.617.  Google Scholar

[25]

B. Q. Yuan, On regularity criteria of weak solutions to the micropolar fluid equations in Lorentz space, Amer. Math. Soc., 138 (2010), 2025-2036. doi: 10.1090/S0002-9939-10-10232-9.  Google Scholar

[26]

B. Q. Yuan, Regularity of weak solutions to magneto-micropolar equations, Acta Math. Sci., 30 (2010), 1469-1480. doi: 10.1016/S0252-9602(10)60139-7.  Google Scholar

[27]

H. Zhang, Logarithmically improved regularity criterion for the 3D micropolar fluid equations, Int. J. Appl. Anal., (2014), Art. ID 386269, 6 pp. doi: 10.1155/2014/386269.  Google Scholar

[28]

Y. Zhou, A new regularity criteria for weak solutions to the Navier-Stokes equations, J. Math. Pures Appl., 84 (2005), 1496-1514. doi: 10.1016/j.matpur.2005.07.003.  Google Scholar

[29]

Y. Zhou and M. Pokorny, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107. doi: 10.1088/0951-7715/23/5/004.  Google Scholar

show all references

References:
[1]

L. C. Berselli, On a regularity criterion for the solutions to 3D Navier-Stokes equations, Differential Integral Equations, 15 (2002), 1129-1137.  Google Scholar

[2]

H. Bahouri, R. Danchin and J. Y. Chemin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer Heidelberg Dordrecht London New York, Springer-Verlag Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[3]

J. Bergh and J. Löfström, Inerpolation Spaces, an Introduction, Springer-Verlag, New York, 1976.  Google Scholar

[4]

Q. L. Chen and C. X. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differential Equations, 252 (2012), 2698-2724. doi: 10.1016/j.jde.2011.09.035.  Google Scholar

[5]

C. S. Cao and J. H. 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

[6]

B. Q. Dong and Z. M. Chen, Regularity criteria of weak solutions to the three-dimensional micropolar flows, J. Math. Phys., 50 (2009), 103525, 13 pp. doi: 10.1063/1.3245862.  Google Scholar

[7]

B. Q. Dong and W. Zhang, On the regularity criterion for the three-dimensional micropolar flows in Besov spaces, Nonlinear Anal., 73 (2010), 2334-2341. doi: 10.1016/j.na.2010.06.029.  Google Scholar

[8]

B. Q. Dong and Z. F. Zhang, The BKM criterion for the 3D Navier-Stokes equations via two velocity components, Nonlinear Anal., 11 (2010), 2415-2421. doi: 10.1016/j.nonrwa.2009.07.013.  Google Scholar

[9]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  Google Scholar

[10]

D. Y. Fang and C. Y. Qian, Regularity criteria for 3D Navier-Stokes equations in Besov space, Comm.Pura Appl, 13 (2014), 585-603. doi: 10.3934/cpaa.2014.13.585.  Google Scholar

[11]

G. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Int. J. Eng. Sci., 15 (1977), 105-108. doi: 10.1016/0020-7225(77)90025-8.  Google Scholar

[12]

S. Gala, On regularity criteria for the three-dimensional micropolar fluid equations in the critical Morrey-Campanato space, Nonlinear Anal., 12 (2011), 2142-2150. doi: 10.1016/j.nonrwa.2010.12.028.  Google Scholar

[13]

I. Kukavica and M. Zinae, Navier-Stokes equation with regularity in one direction, J. Math. Phys., 48 (2007), 065203, 10 pp. doi: 10.1063/1.2395919.  Google Scholar

[14]

O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids, Gorden Brech, New York, 1969. Google Scholar

[15]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hill/CRC, 2002. doi: 10.1201/9781420035674.  Google Scholar

[16]

P. L. Lions and Lions, Mathematical Topics in Fluid Mechanics, Oxford University Press Inc. New York, 1996.  Google Scholar

[17]

G. Lukaszewicz, Micropolar Fluids, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

[18]

G. Lukaszewicz, On nonstationary flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat. (5), 12 (1988), 83-97.  Google Scholar

[19]

G. Lukaszewicz, On the existence, uniqueness and asymptotic properties for solutions of flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat. (5), 13 (1989), 105-120.  Google Scholar

[20]

E. Ortega-Torres and M. Rojas-Medar, On the regularity for solutions of the micropolar fluid equations, Rend. Semin. Mat. Univ. Padova, 122 (2009), 27-37. doi: 10.4171/RSMUP/122-3.  Google Scholar

[21]

M. A. Rojas-Medar and J. L. Boldrini, Magneto-micropolar fluid motion: Existence of weak solutions, Rev. Mat. Complut., 11 (1998), 443-460. doi: 10.5209/rev_REMA.1998.v11.n2.17276.  Google Scholar

[22]

Z. Skalák, Criteria for the regularity of the solutions to the Navier-Stokes equations based on the velocity gradient, Nonlinear Anal., 118 (2015), 1-21. doi: 10.1016/j.na.2015.01.011.  Google Scholar

[23]

Y. X. Wang and H. J. Zhao, Logarithmically improved blow up criterion for smooths solution to the 3D Micropolar Fluid equations, J. Appl. Math., (2012), Art. ID 541203, 13 pp. doi: 10.1016/j.nonrwa.2011.12.018.  Google Scholar

[24]

N. Yamaguchi, Existence of global strong solution to the micropolar fluid system in a bounded domain, Methods Appl. Sci., 28 (2005), 1507-1526. doi: 10.1002/mma.617.  Google Scholar

[25]

B. Q. Yuan, On regularity criteria of weak solutions to the micropolar fluid equations in Lorentz space, Amer. Math. Soc., 138 (2010), 2025-2036. doi: 10.1090/S0002-9939-10-10232-9.  Google Scholar

[26]

B. Q. Yuan, Regularity of weak solutions to magneto-micropolar equations, Acta Math. Sci., 30 (2010), 1469-1480. doi: 10.1016/S0252-9602(10)60139-7.  Google Scholar

[27]

H. Zhang, Logarithmically improved regularity criterion for the 3D micropolar fluid equations, Int. J. Appl. Anal., (2014), Art. ID 386269, 6 pp. doi: 10.1155/2014/386269.  Google Scholar

[28]

Y. Zhou, A new regularity criteria for weak solutions to the Navier-Stokes equations, J. Math. Pures Appl., 84 (2005), 1496-1514. doi: 10.1016/j.matpur.2005.07.003.  Google Scholar

[29]

Y. Zhou and M. Pokorny, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107. doi: 10.1088/0951-7715/23/5/004.  Google Scholar

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