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

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

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
Related Papers

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.

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.
[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.

[1]	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
[2]	Lei Zhang, Bin Liu. Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2655-2685. doi: 10.3934/dcds.2018112
[3]	Alejandro Sarria. Global estimates and blow-up criteria for the generalized Hunter-Saxton system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 641-673. doi: 10.3934/dcdsb.2015.20.641
[4]	Jihoon Lee. Scaling invariant blow-up criteria for simplified versions of Ericksen-Leslie system. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 381-388. doi: 10.3934/dcdss.2015.8.381
[5]	Wenjing Zhao. Local well-posedness and blow-up criteria of magneto-viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4637-4655. doi: 10.3934/dcds.2018203
[6]	Yan Jia, Xingwei Zhang, Bo-Qing Dong. Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion. Communications on Pure & Applied Analysis, 2013, 12 (2) : 923-937. doi: 10.3934/cpaa.2013.12.923
[7]	Hua Qiu. Regularity criteria of smooth solution to the incompressible viscoelastic flow. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2873-2888. doi: 10.3934/cpaa.2013.12.2873
[8]	Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225
[9]	Yukihiro Seki. A remark on blow-up at space infinity. Conference Publications, 2009, 2009 (Special) : 691-696. doi: 10.3934/proc.2009.2009.691
[10]	Jacek Banasiak. Blow-up of solutions to some coagulation and fragmentation equations with growth. Conference Publications, 2011, 2011 (Special) : 126-134. doi: 10.3934/proc.2011.2011.126
[11]	Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733
[12]	Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771
[13]	Marius Ghergu, Vicenţiu Rădulescu. Nonradial blow-up solutions of sublinear elliptic equations with gradient term. Communications on Pure & Applied Analysis, 2004, 3 (3) : 465-474. doi: 10.3934/cpaa.2004.3.465
[14]	Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54
[15]	Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881
[16]	Zhifu Xie. General uniqueness results and examples for blow-up solutions of elliptic equations. Conference Publications, 2009, 2009 (Special) : 828-837. doi: 10.3934/proc.2009.2009.828
[17]	Yu-Zhu Wang, Weibing Zuo. On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1327-1336. doi: 10.3934/cpaa.2014.13.1327
[18]	Dongho Chae. On the blow-up problem for the Euler equations and the Liouville type results in the fluid equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1139-1150. doi: 10.3934/dcdss.2013.6.1139
[19]	Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831
[20]	C. Y. Chan. Recent advances in quenching and blow-up of solutions. Conference Publications, 2001, 2001 (Special) : 88-95. doi: 10.3934/proc.2001.2001.88

2017 Impact Factor: 0.561

American Institute of Mathematical Sciences