AIMS Mathematics, 2020, 5(1): 359-375. doi: 10.3934/math.2020024.

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

The anisotropic integrability logarithmic regularity criterion to the 3D micropolar fluid equations

1 Department of Mathematical Science, Faculty of Applied Science, Umm Alqura University, P. O. Box 14035, Makkah 21955, Saudi Arabia
2 Department of Mathematics, ENS of Mostaganem, Box 227, Mostaganem 27000, Algeria
3 Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria, 6 95125 Catania - Italy
4 Department of Mathematical Education, Andong National University, Andong, Gyeongsangbuk-do, 36729, Korea (Republic of)
5 RUDN University, 6 Miklukho - Maklay St, Moscow, 117198, Russia

The aim of this paper is to establish the regularity criterion of weak solutions to the 3D micropolar fluid equations by one directional derivative of the pressure in anisotropic Lebesgue spaces. We improve the regularity criterion for weak solutions previously given by Jia, Zhang and Dong in [21].
  Figure/Table
  Supplementary
  Article Metrics

Keywords micropolar fluid equations; regularity criterion; anisotropic Lebesgue spaces; a priori estimates

Citation: Ahmad Mohammad Alghamdi, Sadek Gala, Jae-Myoung Kim, Maria Alessandra Ragusa. The anisotropic integrability logarithmic regularity criterion to the 3D micropolar fluid equations. AIMS Mathematics, 2020, 5(1): 359-375. doi: 10.3934/math.2020024

References

  • 1. J. Chen, Z. M. Chen and B. Q. Dong, Uniform attractors of non-homogeneous micropolar fluid flows in non-smooth domains, Nonlinearity, 20 (2007), 1619-1635.    
  • 2. Q. Chen and C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differ. Equations, 252 (2012), 2698-2724.    
  • 3. Z. M. Chen and W. Price, Decay estimates of linearized micropolar fluid flows in R3 space with applications to L3-strong solutions, Int. J. Eng. Sci., 44 (2006), 859-873.    
  • 4. B. Q. Dong and Z. M. Chen, Regularity criteria of weak solutions to the three-dimensional micropolar flows, J. Math. Phys., 50 (2009), 103525.
  • 5. B. Q. Dong and Z. M. Chen, Global attractors of two-dimensional micropolar fluid flows in some unbounded domains, Appl. Math. Comput., 182 (2006), 610-620.
  • 6. B. Q. Dong and Z. M. Chen, On upper and lower bounds of higher order derivatives for solutions to the 2D micropolar fluid equations, J. Math. Anal. Appl., 334 (2007), 1386-1399.    
  • 7. B. Q. Dong and Z. M. Chen, Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows, Discrete and Continuous Dynamics Systems, 23 (2009), 765-784.
  • 8. B. Q. Dong and Z. Zhang, Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differ. Equations, 249 (2010), 200-213    
  • 9. B. Q. Dong and W. Zhang, On the regularity criterion for the 3D micropolar fluid flows in Besov spaces, Nonlinear Analysis: Theory, Methods & Applications, 73 (2010), 2334-2341.
  • 10. B. Q. Dong, Y. Jia and Z. M. Chen, Pressure regularity criteria of the three-dimensional micropolar fluid flows, Math. Meth. Appl. Sci., 34 (2011), 595-606.    
  • 11. A. C. Eringen, Theory of micropolar fluids, Journal of Mathematics and Mechanics, 16 (1966), 1-18.
  • 12. J. Fan, X. Jia and Y. Zhou, A logarithmic regularity criterion for 3D Navier-Stokes system in a bounded domain, Appl. Math., 64 (2019), 397-407.
  • 13. J. Fan, Y. Fukumoto and Y. Zhou, Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations, Kinet. Relat. Models, 6 (2013), 545-556.    
  • 14. J. Fan, S. Jiang, G. Nakamura, et al. Logarithmically improved regularity criteria for the NavierStokes and MHD equations, J. Math. Fluid Mech., 13 (2011), 557-571.    
  • 15. S. Gala, On regularity criteria for the three-dimensional micropolar fluid equations in the critical Morrey-Campanato space, Nonlinear Analysis : Real World Applications, 12 (2011), 2142-2150.    
  • 16. S. Gala and M. A. Ragusa, A regularity criterion for 3D micropolar fluid flows in terms of one partial derivative of the velocity, Annales Polonici Mathematici, 116 (2016), 217-228.
  • 17. S. Gala and J. Yan, Two regularity criteria via the logarithmic of the weak solutions to the micropolar fluid equations, J. Partial Differ. Equ., 25 (2012), 32-40.    
  • 18. S. Gala, A remark on the logarithmically improved regularity criterion for the micropolar fluid equations in terms of the pressure, Math. Meth. Appl. Sci., 34 (2011), 1945-1953.    
  • 19. G. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of micropolar fluid equations, Int. J. Eng. Sci., 15 (1977), 105-108.    
  • 20. Y. Jia, W. Zhang and B. Dong, Remarks on the regularity criterion of the 3D micropolar fluid flows in terms of the pressure, Appl. Math. Lett., 24 (2011), 199-203.    
  • 21. Y. Jia, W. Zhang and B. Dong, Logarithmical regularity criteria of the three-dimensional micropolar fluid equations in terms of the pressure, Abstr. Appl. Anal., 2012 (2012), 1-13.
  • 22. X. Jia and Y. Zhou, A new regularity criterion for the 3D incompressible MHD equations in terms of one component of the gradient of pressure, J. Math. Anal. Appl., 396 (2012), 345-350.    
  • 23. E. Hopf, Über die anfangswertaufgabe fur die hydrodynamischen grundgleichungen. Erhard Schmidt zu seinem 75. Geburtstag gewidmet, Math. Nachr., 4 (1950), 213-231.
  • 24. J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 183-248.
  • 25. Q. Liu and G. Dai, On the 3D Navier-Stokes equations with regularity in pressure, J. Math. Anal. Appl., 458 (2018), 497-507.    
  • 26. G. Lukaszewicz, Micropolar fluids. Theory and applications, Modeling and Simulation in Science, Engineering and Technology, Birkhauser, Boston, MA, 1999.
  • 27. M. A. Rojas-Medar, Magneto-micropolar fluid motion: existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319.    
  • 28. S. Popel, A. Regirer and P. Usick, A continuum model of blood flow, Biorheology, 11 (1974), 427-437.    
  • 29. R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, The Netherlands, 1977.
  • 30. M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat., 18 (1969), 3-24.
  • 31. B. Yuan, On the regularity criteria of weak solutions to the micropolar fluid equations in Lorentz space, P. Am. Math. Soc., 138 (2010), 2025-2036.    
  • 32. J. Yuan, Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations, Math. Meth. Appl. Sci., 31 (2008), 1113-1130.    
  • 33. Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708.

 

Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved