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doi: 10.3934/era.2020083

Regularity criteria for weak solutions of the Magneto-micropolar equations

Jens Lorenz 1, , Wilberclay G. Melo 2,, and  Suelen C. P. de Souza 2,

1. 

Department of Mathematics and Statistics, The University of New Mexico (UNM), Albuquerque, NM 87131, United States of America
2. 

Departamento de Matemática, Universidade Federal de Segipe (UFS), São Cristóvão, SE 49100-000, Brazil

* Corresponding author: Wilberclay G. Melo

Received  January 2020 Revised  April 2020 Published  August 2020

Fund Project: The first author is supported by NSF grant DMS-1148801

In this paper, we show that a weak solution $ (\mathbf{u},\mathbf{w},\mathbf{b})(\cdot,t) $ of the magneto-micropolar equations, defined in $ [0,T) $, which satisfies $ \nabla u_3, \nabla_{h} \mathbf{w}, \nabla_{h} \mathbf{b} $ $ \in L^{\frac{32}{7}}(0,T; $ $ L^2(\mathbb{R}^3)) $ or $ \partial_3 u_3, \partial_3 \mathbf{w}, \partial_3 \mathbf{b} \in L^{\infty}(0,T;L^2(\mathbb{R}^3)) $, is regular in $ \mathbb{R}^3\times(0,T) $ and can be extended as a $ C^\infty $ solution beyond $ T $.

Keywords: Magneto-micropolar equations, weak solutions, regularity criteria.
Mathematics Subject Classification: Primary: 76W05, 35D30, 35B65, 35Q30.
Citation: Jens Lorenz, Wilberclay G. Melo, Suelen C. P. de Souza. Regularity criteria for weak solutions of the Magneto-micropolar equations. Electronic Research Archive, doi: 10.3934/era.2020083
References:
[1]

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E. E. Ortega-Torres and M. A. Rojas-Medar, Magneto-micropolar fluid motion: Global existence of strong solutions, Abstr. Appl. Anal., 4 (1999), 109-125.  doi: 10.1155/S1085337599000287.  Google Scholar

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P. Penel and M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Appl. Math., 49 (2004), 483-493.  doi: 10.1023/B:APOM.0000048124.64244.7e.  Google Scholar

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M. A. Rojas-Medar, Magneto-micropolar fluid motion: Existence and uniqueness of strong solutions, Math. Nachr., 188 (1997), 301-319.  doi: 10.1002/mana.19971880116.  Google Scholar

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Z. Skalák, A note on the regularity of the solutions to the Navier-Stokes equations via the gradient of one velocity component, J. Math. Phys., 55 (2014), 121506, 6pp. doi: 10.1063/1.4904836.  Google Scholar

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Z. Skalák, On the regularity of the solutions to the Navier-Stokes equations via the gradient of one velocity component, Nonlinear Anal., 104 (2014), 84-89.  doi: 10.1016/j.na.2014.03.018.  Google Scholar

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Y. Wang and L. Gu, Global regularity of 3D magneto-micropolar fluid equations,, Appl. Math. Lett., 99 (2020), 105980, 9 pp. doi: 10.1016/j.aml.2019.07.011.  Google Scholar

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

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Z. Zhang, A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component, Commun. Pure Appl. Anal., 12 (2013), 117-124.  doi: 10.3934/cpaa.2013.12.117.  Google Scholar

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Z. Zhang, An almost Serrin-type regularity criterion for the Navier-Stokes equations involving the gradient of one velocity component, Z. Angew. Math. Phys., 66 (2015), 1707-1715.  doi: 10.1007/s00033-015-0500-7.  Google Scholar

[21]

Z. Zhang, Regularity criteria for the 3D MHD equations involving one current density and the gradient of one velocity component, Nonlinear Anal., 115 (2015), 41-49.  doi: 10.1016/j.na.2014.12.003.  Google Scholar

[22]

Z. Zhang and X. Yang, A note on the regularity criterion for the 3D Navier-Stokes equations via the gradient of one velocity component, J. Math. Anal. Appl., 432 (2015), 603-611.  doi: 10.1016/j.jmaa.2015.06.050.  Google Scholar

[23]

Z. Zhang and X. Yang, On the regularity criterion for the Navier-Stokes equations involving the diagonal entry of the velocity gradient, Nonlinear Anal., 122 (2015), 169-175.  doi: 10.1016/j.na.2015.04.005.  Google Scholar

[24]

Z. Zhang, Z.-A. Yao, P. Li, C. Guo and M. Lu, Two new regularity criteria for the $3D$ Navier-Stokes equations via two entries of the velocity gradient tensor, Acta Appl. Math., 123 (2013), 43-52.  doi: 10.1007/s10440-012-9712-4.  Google Scholar

[25]

Z. Zhang, D. Zhong and L. Hu, A new regularity criterion for the $3D$ Navier-Stokes equations via two entries of the velocity gradient tensor, Acta Appl. Math., 129 (2014), 175-181.  doi: 10.1007/s10440-013-9834-3.  Google Scholar

[26]

X. Zheng, A regularity criterion for the tridimensional Navier-Stokes equations in term of one velocity component, J. Differential Equations, 256 (2014), 283-309.  doi: 10.1016/j.jde.2013.09.002.  Google Scholar

[27]

Y. Zhou, A new regularity criterion for the Navier-Stokes equations in terms of the gradient of one velocity component, Methods Appl. Anal., 9 (2002), 563-578.  doi: 10.4310/MAA.2002.v9.n4.a5.  Google Scholar

[28]

Y. Zhou and M. Pokorný, On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component, J. Math. Phys., 50 (2009), 123514, 11pp. doi: 10.1063/1.3268589.  Google Scholar

[29]

Y. Zhou and M. Pokorný, 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]

C. Cao and E. S. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal., 202 (2011), 919-932.  doi: 10.1007/s00205-011-0439-6.  Google Scholar

[2]

C. Cao and J. Wu, Two regularity criteria for the $3D$ MHD equations, J. Differ. Equ., 248 (2010), 2263-2274.  doi: 10.1016/j.jde.2009.09.020.  Google Scholar

[3]

C. He, Regularity for solutions to the Navier-Stokes equations with one velocity component regular, Electron. J. Differencial Equations, 29 (2002), 13pp.  Google Scholar

[4]

X. Jia and Y. Zhou, Remarks on regularity criteria for the Navier-Stokes equations via one velocity component, Nonlinear Anal. Real World Appl., 15 (2014), 239-245.  doi: 10.1016/j.nonrwa.2013.08.002.  Google Scholar

[5]

I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 453-469.  doi: 10.1088/0951-7715/19/2/012.  Google Scholar

[6]

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

[7]

J. Neustupa, A. Novotný and P. Penel, An interior regularity of a weak solution to the Navier-Stokes equations in dependence on one component of velocity, topics in mathematical fluid mechanics, Quad. Mat., 10 (2002), 163-183.   Google Scholar

[8]

J. Neustupa and P. Penel, Regularity of a suitable weak solution to the Navier-Stokes equations as a consequence of regularity of one velocity component, Applied Nonlinear Analysis., Kluwer/Plenum, New York, (1999), 391–402.  Google Scholar

[9]

E. E. Ortega-Torres and M. A. Rojas-Medar, Magneto-micropolar fluid motion: Global existence of strong solutions, Abstr. Appl. Anal., 4 (1999), 109-125.  doi: 10.1155/S1085337599000287.  Google Scholar

[10]

P. Penel and M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Appl. Math., 49 (2004), 483-493.  doi: 10.1023/B:APOM.0000048124.64244.7e.  Google Scholar

[11]

M. A. Rojas-Medar, Magneto-micropolar fluid motion: Existence and uniqueness of strong solutions, Math. Nachr., 188 (1997), 301-319.  doi: 10.1002/mana.19971880116.  Google Scholar

[12]

Z. Skalák, A note on the regularity of the solutions to the Navier-Stokes equations via the gradient of one velocity component, J. Math. Phys., 55 (2014), 121506, 6pp. doi: 10.1063/1.4904836.  Google Scholar

[13]

Z. Skalák, On the regularity of the solutions to the Navier-Stokes equations via the gradient of one velocity component, Nonlinear Anal., 104 (2014), 84-89.  doi: 10.1016/j.na.2014.03.018.  Google Scholar

[14]

Y. Wang, Regularity criterion for a weak solution to the three-dimensional magneto-micropolar fluid equations, Bound. Value Probl., 2013 (2013), 12pp. doi: 10.1186/1687-2770-2013-58.  Google Scholar

[15]

F. Wang, On global regularity of incompressile MHD equations in $\Bbb R^3$, J. Math. Anal. Appl., 454 (2017), 936-941.  doi: 10.1016/j.jmaa.2017.05.045.  Google Scholar

[16]

F. Wang and K. Wang, Global existence of 3D MHD equations with mixed partial dissipation and magnetic diffusion, Nonlinear Anal. Real World Appl., 14 (2013), 526-535.  doi: 10.1016/j.nonrwa.2012.07.013.  Google Scholar

[17]

Y. Wang and L. Gu, Global regularity of 3D magneto-micropolar fluid equations,, Appl. Math. Lett., 99 (2020), 105980, 9 pp. doi: 10.1016/j.aml.2019.07.011.  Google Scholar

[18]

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

[19]

Z. Zhang, A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component, Commun. Pure Appl. Anal., 12 (2013), 117-124.  doi: 10.3934/cpaa.2013.12.117.  Google Scholar

[20]

Z. Zhang, An almost Serrin-type regularity criterion for the Navier-Stokes equations involving the gradient of one velocity component, Z. Angew. Math. Phys., 66 (2015), 1707-1715.  doi: 10.1007/s00033-015-0500-7.  Google Scholar

[21]

Z. Zhang, Regularity criteria for the 3D MHD equations involving one current density and the gradient of one velocity component, Nonlinear Anal., 115 (2015), 41-49.  doi: 10.1016/j.na.2014.12.003.  Google Scholar

[22]

Z. Zhang and X. Yang, A note on the regularity criterion for the 3D Navier-Stokes equations via the gradient of one velocity component, J. Math. Anal. Appl., 432 (2015), 603-611.  doi: 10.1016/j.jmaa.2015.06.050.  Google Scholar

[23]

Z. Zhang and X. Yang, On the regularity criterion for the Navier-Stokes equations involving the diagonal entry of the velocity gradient, Nonlinear Anal., 122 (2015), 169-175.  doi: 10.1016/j.na.2015.04.005.  Google Scholar

[24]

Z. Zhang, Z.-A. Yao, P. Li, C. Guo and M. Lu, Two new regularity criteria for the $3D$ Navier-Stokes equations via two entries of the velocity gradient tensor, Acta Appl. Math., 123 (2013), 43-52.  doi: 10.1007/s10440-012-9712-4.  Google Scholar

[25]

Z. Zhang, D. Zhong and L. Hu, A new regularity criterion for the $3D$ Navier-Stokes equations via two entries of the velocity gradient tensor, Acta Appl. Math., 129 (2014), 175-181.  doi: 10.1007/s10440-013-9834-3.  Google Scholar

[26]

X. Zheng, A regularity criterion for the tridimensional Navier-Stokes equations in term of one velocity component, J. Differential Equations, 256 (2014), 283-309.  doi: 10.1016/j.jde.2013.09.002.  Google Scholar

[27]

Y. Zhou, A new regularity criterion for the Navier-Stokes equations in terms of the gradient of one velocity component, Methods Appl. Anal., 9 (2002), 563-578.  doi: 10.4310/MAA.2002.v9.n4.a5.  Google Scholar

[28]

Y. Zhou and M. Pokorný, On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component, J. Math. Phys., 50 (2009), 123514, 11pp. doi: 10.1063/1.3268589.  Google Scholar

[29]

Y. Zhou and M. Pokorný, 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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