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Regularity criteria for weak solutions of the Magneto-micropolar equations
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 |
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 $.
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. |
[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. |
[3] |
C. He, Regularity for solutions to the Navier-Stokes equations with one velocity component regular, Electron. J. Differencial Equations, 29 (2002), 13pp. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[3] |
C. He, Regularity for solutions to the Navier-Stokes equations with one velocity component regular, Electron. J. Differencial Equations, 29 (2002), 13pp. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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