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

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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

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

Abstract
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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.
[2]	H. Bahouri, R. Danchin and J. Y. Chemin, Fourier Analysis and Nonlinear Partial Differential Equations,, Springer Heidelberg Dordrecht London New York, (2011). doi: 10.1007/978-3-642-16830-7.
[3]	J. Bergh and J. Löfström, Inerpolation Spaces, an Introduction,, Springer-Verlag, (1976).
[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. doi: 10.1016/j.jde.2011.09.035.
[5]	C. S. Cao and J. H. Wu, Two regularity criteria for the 3D MHD equations,, J. Differential Equations, 248 (2010), 2263. doi: 10.1016/j.jde.2009.09.020.
[6]	B. Q. Dong and Z. M. Chen, Regularity criteria of weak solutions to the three-dimensional micropolar flows,, J. Math. Phys., 50 (2009). doi: 10.1063/1.3245862.
[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. doi: 10.1016/j.na.2010.06.029.
[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. doi: 10.1016/j.nonrwa.2009.07.013.
[9]	A. C. Eringen, Theory of micropolar fluids,, J. Math. Mech., 16 (1966), 1.
[10]	D. Y. Fang and C. Y. Qian, Regularity criteria for 3D Navier-Stokes equations in Besov space,, Comm.Pura Appl, 13 (2014), 585. doi: 10.3934/cpaa.2014.13.585.
[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. doi: 10.1016/0020-7225(77)90025-8.
[12]	S. Gala, On regularity criteria for the three-dimensional micropolar fluid equations in the critical Morrey-Campanato space,, Nonlinear Anal., 12 (2011), 2142. doi: 10.1016/j.nonrwa.2010.12.028.
[13]	I. Kukavica and M. Zinae, Navier-Stokes equation with regularity in one direction,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2395919.
[14]	O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids,, Gorden Brech, (1969).
[15]	P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem,, Chapman & Hill/CRC, (2002). doi: 10.1201/9781420035674.
[16]	P. L. Lions and Lions, Mathematical Topics in Fluid Mechanics,, Oxford University Press Inc. New York, (1996).
[17]	G. Lukaszewicz, Micropolar Fluids,, Birkhäuser, (1999). doi: 10.1007/978-1-4612-0641-5.
[18]	G. Lukaszewicz, On nonstationary flows of asymmetric fluids,, Rend. Accad. Naz. Sci. XL Mem. Mat. (5), 12 (1988), 83.
[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.
[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. doi: 10.4171/RSMUP/122-3.
[21]	M. A. Rojas-Medar and J. L. Boldrini, Magneto-micropolar fluid motion: Existence of weak solutions,, Rev. Mat. Complut., 11 (1998), 443. doi: 10.5209/rev_REMA.1998.v11.n2.17276.
[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. doi: 10.1016/j.na.2015.01.011.
[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). doi: 10.1016/j.nonrwa.2011.12.018.
[24]	N. Yamaguchi, Existence of global strong solution to the micropolar fluid system in a bounded domain,, Methods Appl. Sci., 28 (2005), 1507. doi: 10.1002/mma.617.
[25]	B. Q. Yuan, On regularity criteria of weak solutions to the micropolar fluid equations in Lorentz space,, Amer. Math. Soc., 138 (2010), 2025. doi: 10.1090/S0002-9939-10-10232-9.
[26]	B. Q. Yuan, Regularity of weak solutions to magneto-micropolar equations,, Acta Math. Sci., 30 (2010), 1469. doi: 10.1016/S0252-9602(10)60139-7.
[27]	H. Zhang, Logarithmically improved regularity criterion for the 3D micropolar fluid equations,, Int. J. Appl. Anal., (2014). doi: 10.1155/2014/386269.
[28]	Y. Zhou, A new regularity criteria for weak solutions to the Navier-Stokes equations,, J. Math. Pures Appl., 84 (2005), 1496. doi: 10.1016/j.matpur.2005.07.003.
[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. doi: 10.1088/0951-7715/23/5/004.

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References:

[1]	L. C. Berselli, On a regularity criterion for the solutions to 3D Navier-Stokes equations,, Differential Integral Equations, 15 (2002), 1129.
[2]	H. Bahouri, R. Danchin and J. Y. Chemin, Fourier Analysis and Nonlinear Partial Differential Equations,, Springer Heidelberg Dordrecht London New York, (2011). doi: 10.1007/978-3-642-16830-7.
[3]	J. Bergh and J. Löfström, Inerpolation Spaces, an Introduction,, Springer-Verlag, (1976).
[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. doi: 10.1016/j.jde.2011.09.035.
[5]	C. S. Cao and J. H. Wu, Two regularity criteria for the 3D MHD equations,, J. Differential Equations, 248 (2010), 2263. doi: 10.1016/j.jde.2009.09.020.
[6]	B. Q. Dong and Z. M. Chen, Regularity criteria of weak solutions to the three-dimensional micropolar flows,, J. Math. Phys., 50 (2009). doi: 10.1063/1.3245862.
[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. doi: 10.1016/j.na.2010.06.029.
[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. doi: 10.1016/j.nonrwa.2009.07.013.
[9]	A. C. Eringen, Theory of micropolar fluids,, J. Math. Mech., 16 (1966), 1.
[10]	D. Y. Fang and C. Y. Qian, Regularity criteria for 3D Navier-Stokes equations in Besov space,, Comm.Pura Appl, 13 (2014), 585. doi: 10.3934/cpaa.2014.13.585.
[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. doi: 10.1016/0020-7225(77)90025-8.
[12]	S. Gala, On regularity criteria for the three-dimensional micropolar fluid equations in the critical Morrey-Campanato space,, Nonlinear Anal., 12 (2011), 2142. doi: 10.1016/j.nonrwa.2010.12.028.
[13]	I. Kukavica and M. Zinae, Navier-Stokes equation with regularity in one direction,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2395919.
[14]	O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids,, Gorden Brech, (1969).
[15]	P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem,, Chapman & Hill/CRC, (2002). doi: 10.1201/9781420035674.
[16]	P. L. Lions and Lions, Mathematical Topics in Fluid Mechanics,, Oxford University Press Inc. New York, (1996).
[17]	G. Lukaszewicz, Micropolar Fluids,, Birkhäuser, (1999). doi: 10.1007/978-1-4612-0641-5.
[18]	G. Lukaszewicz, On nonstationary flows of asymmetric fluids,, Rend. Accad. Naz. Sci. XL Mem. Mat. (5), 12 (1988), 83.
[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.
[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. doi: 10.4171/RSMUP/122-3.
[21]	M. A. Rojas-Medar and J. L. Boldrini, Magneto-micropolar fluid motion: Existence of weak solutions,, Rev. Mat. Complut., 11 (1998), 443. doi: 10.5209/rev_REMA.1998.v11.n2.17276.
[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. doi: 10.1016/j.na.2015.01.011.
[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). doi: 10.1016/j.nonrwa.2011.12.018.
[24]	N. Yamaguchi, Existence of global strong solution to the micropolar fluid system in a bounded domain,, Methods Appl. Sci., 28 (2005), 1507. doi: 10.1002/mma.617.
[25]	B. Q. Yuan, On regularity criteria of weak solutions to the micropolar fluid equations in Lorentz space,, Amer. Math. Soc., 138 (2010), 2025. doi: 10.1090/S0002-9939-10-10232-9.
[26]	B. Q. Yuan, Regularity of weak solutions to magneto-micropolar equations,, Acta Math. Sci., 30 (2010), 1469. doi: 10.1016/S0252-9602(10)60139-7.
[27]	H. Zhang, Logarithmically improved regularity criterion for the 3D micropolar fluid equations,, Int. J. Appl. Anal., (2014). doi: 10.1155/2014/386269.
[28]	Y. Zhou, A new regularity criteria for weak solutions to the Navier-Stokes equations,, J. Math. Pures Appl., 84 (2005), 1496. doi: 10.1016/j.matpur.2005.07.003.
[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. doi: 10.1088/0951-7715/23/5/004.

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