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

Baoquan Yuan; Xiao  Li

doi:10.3934/dcdss.2016090

Article Contents

2016, Volume 9, Issue 6: 2167-2179. Doi: 10.3934/dcdss.2016090

This issue Previous Article Wave breaking and persistent decay of solution to a shallow water wave equation Next Article Decay estimates with sharp rates of global solutions of nonlinear systems of fluid dynamics equations

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

Received: June 30, 2015

Revised: August 31, 2016

Published: November 2016

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35Q35, 35B65; Secondary: 76N10.

Citation:

Related Papers

Cited by

References

[1]	L. C. Berselli, On a regularity criterion for the solutions to 3D Navier-Stokes equations, Differential Integral Equations, 15 (2002), 1129-1137.
[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.
[3]	J. Bergh and J. Löfström, Inerpolation Spaces, an Introduction, Springer-Verlag, New York, 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), 2698-2724.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), 2263-2274.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), 103525, 13 pp.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), 2334-2341.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), 2415-2421.doi: 10.1016/j.nonrwa.2009.07.013.
[9]	A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.
[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.
[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.
[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.
[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.
[14]	O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids, Gorden Brech, New York, 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, Boston, 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), 83-97.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.