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 $.
Citation: Jens Lorenz, Wilberclay G. Melo, Suelen C. P. de Souza. Regularity criteria for weak solutions of the Magneto-micropolar equations[J]. Electronic Research Archive, 2021, 29(1): 1625-1639. doi: 10.3934/era.2020083
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 $.
[1] |
Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor. Arch. Ration. Mech. Anal. (2011) 202: 919-932. ![]() |
[2] |
Two regularity criteria for the $3D$ MHD equations. J. Differ. Equ. (2010) 248: 2263-2274. ![]() |
[3] | C. He, Regularity for solutions to the Navier-Stokes equations with one velocity component regular, Electron. J. Differencial Equations, 29 (2002), 13pp. |
[4] |
Remarks on regularity criteria for the Navier-Stokes equations via one velocity component. Nonlinear Anal. Real World Appl. (2014) 15: 239-245. ![]() |
[5] |
One component regularity for the Navier-Stokes equations. Nonlinearity (2006) 19: 453-469. ![]() |
[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] | 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. (2002) 10: 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] |
Magneto-micropolar fluid motion: Global existence of strong solutions. Abstr. Appl. Anal. (1999) 4: 109-125. ![]() |
[10] |
Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity. Appl. Math. (2004) 49: 483-493. ![]() |
[11] |
Magneto-micropolar fluid motion: Existence and uniqueness of strong solutions. Math. Nachr. (1997) 188: 301-319. ![]() |
[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] |
On the regularity of the solutions to the Navier-Stokes equations via the gradient of one velocity component. Nonlinear Anal. (2014) 104: 84-89. ![]() |
[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] |
On global regularity of incompressile MHD equations in $\Bbb R^3$. J. Math. Anal. Appl. (2017) 454: 936-941. ![]() |
[16] |
Global existence of 3D MHD equations with mixed partial dissipation and magnetic diffusion. Nonlinear Anal. Real World Appl. (2013) 14: 526-535. ![]() |
[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] |
Regularity of weak solutions to magneto-micropolar fluid equations. Acta Math. Sci. Ser. B (2010) 30: 1469-1480. ![]() |
[19] |
A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component. Commun. Pure Appl. Anal. (2013) 12: 117-124. ![]() |
[20] |
An almost Serrin-type regularity criterion for the Navier-Stokes equations involving the gradient of one velocity component. Z. Angew. Math. Phys. (2015) 66: 1707-1715. ![]() |
[21] |
Regularity criteria for the 3D MHD equations involving one current density and the gradient of one velocity component. Nonlinear Anal. (2015) 115: 41-49. ![]() |
[22] |
A note on the regularity criterion for the 3D Navier-Stokes equations via the gradient of one velocity component. J. Math. Anal. Appl. (2015) 432: 603-611. ![]() |
[23] |
On the regularity criterion for the Navier-Stokes equations involving the diagonal entry of the velocity gradient. Nonlinear Anal. (2015) 122: 169-175. ![]() |
[24] |
Two new regularity criteria for the $3D$ Navier-Stokes equations via two entries of the velocity gradient tensor. Acta Appl. Math. (2013) 123: 43-52. ![]() |
[25] |
A new regularity criterion for the $3D$ Navier-Stokes equations via two entries of the velocity gradient tensor. Acta Appl. Math. (2014) 129: 175-181. ![]() |
[26] |
A regularity criterion for the tridimensional Navier-Stokes equations in term of one velocity component. J. Differential Equations (2014) 256: 283-309. ![]() |
[27] |
A new regularity criterion for the Navier-Stokes equations in terms of the gradient of one velocity component. Methods Appl. Anal. (2002) 9: 563-578. ![]() |
[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] |
On the regularity of the solutions of the Navier-Stokes equations via one velocity component. Nonlinearity (2010) 23: 1097-1107. ![]() |