International Journal of Nonlinear Analysis and Applications

Two new regularity criteria for the 3D magneto-micropolar equations in anisotropic Lorentz spaces

Document Type : Research Paper

Authors

1 Department of Mathematics, Quaid-I-Azam University 45320, Islamabad, Pakistan

2 IBADAT International University, 6.7 km Japan Rd, Sihala, Islamabad, Pakistan

Abstract

In this study, we present two new regularity criteria based on pressure and its gradient to the Cauchy problem of the 3D magneto-micropolar system in anisotropic Lorentz spaces.

Keywords

[1] G. Ahmadi and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions, Internat J. Engrg. Sci. 12 (1974), no. 7, 657–663.
[2] K.A. Bekmaganbetov and Y. Toleugazy, On the Order of the trigonometric diameter of the anisotropic Nikol’skiiBesov class in the metric of anisotropic Lorentz spaces, Anal. Math. 45 (2019), 237–247.
[3] C. Berselli and P. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations, Proc. Amer. Math. Soc. 130 (2002), no. 12, 3585–3595.
[4] T. Chen and W. Sun, Iterated weak and weak mixed-norm spaces with applications to geometric inequalities, J. Geom. Anal. 30 (2020), no. 4, 4268–4323.
[5] B. Dong, Y. Jia, and Z. Chen, Pressure regularity criteria of the three-dimensional micropolar fluid flows, Math. Meth. Appl. Sci. 34 (2011), no. 5, 595–606.
[6] H. Duan, On regularity criteria in terms of pressure for the 3D viscous MHD equations, Appl. Anal. 91 (2012), no. 5, 947–952.
[7] A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1996), no. 1, 1–18.
[8] L. Feng-Ping and C. Guang-Xia, Regularity criteria for weak solutions to the 3D magneto-micropolar fluid equations, J. SD. Univ. (Nat Sci.) 50 (2015), 60–67.
[9] G. Gala, Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey–Campanato space, Nonlinear Differ. Equ. Appl. 17 (2010), no. 2, 181—194.
[10] 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), no. 34, 1945–1953.
[11] G.P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Int. J. Eng. Sci. 15 (1977), no. 2, 105–108.
[12] J. Geng, X. Chen, and S. Gala, On regularity criteria for the 3D magneto-micropolar fluid equations in the critical Morrey-Campanato space, Commun. Pure Appl. Anal. 10 (2011), no. 2, 583–592.
[13] Z. Guo, T. Tong, and W. Wang, On regularity of the 3D MHD equations based on one velocity component in anisotropic Lebesgue spaces, Appl. Math. Lett. 120 (2021), 1–7.
[14] Z. Guo and S. Zhang, An optimal regularity criterion for the 3D MHD equations in homogeneous Besov spaces, Math. Meth. Appl. Sci. 44 (2021), no. 2, 2130–2139.
[15] 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), no. 2, 199–203.
[16] X. Jia and Y. Zhou, On Regularity Criteria for the 3D Incompressible MHD Equations Involving One Velocity Component, J. Math. Fluid Mech. 18 (2016), no. 1, 187–206.
[17] Z. Li and P. Niu, New regularity criteria for the 3D magneto-micropolar fluid equations in Lorentz spaces, Math. Meth. Appl. Sci. 44 (2021), no. 7, 6056–6066.
[18] G. Lukaszewicz, Micropolar Fluids: Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, 1st ed., Birkhauser Boston, New York, 1999.
[19] E. Mallea-Zepeda and E.E. Ortega-Torres, Control problem for a magneto-micropolar flow with mixed boundary conditions for the velocity field, J. Dyn. Control Syst. 25 (2019), no. 4, 599–618.
[20] W.G. Melo, The magneto-micropolar equations with periodic boundary conditions: solution properties at potential blow-up times, J. Math. Anal. Appl. 435 (2016), no. 2, 1194–1209.
[21] R. O’Neil, Convolution operators and L(p,q) spaces, Duke Math. J, 30 (1963), no. 1, 129–142.
[22] E.E. Ortega-Torres and M.A. Rojas-Medar, Rojas-Medar, Magneto-micropolar fluid motion: global existence of strong solutions, Abstr. Appl. Anal. 4 (1999), no. 2, 109–125.
[23] E.E. Ortega-Torres and M.A. Rojas-Medar, On the regularity for solutions of the micropolar fluid equations, Rend. Sem. Mat. Padova. 122 (2009), 27–37.
[24] M.A. Ragusa and F. Wu, Regularity criteria for the 3D magneto-hydrodynamics equations in anisotropic Lorentz spaces, Symmetry 13 (2021), no. 4, 1–10.
[25] M.A. Rojas-Medar and J.L. Boldrini, Magneto-micropolar fluid motion: Existence of weak solutions, Rev. Mat. Complut., 11 (1988), no. 2, 443–460.
[26] M.A. Rojas-Medar and M. Loazya, A weak-Lp Prodi-Serrin type regularity criterion for the micropolar fluid equations, J. Math. Phys. 57 (2016), no. 2, 1–6.
[27] P.B. Silva, L. Friz, and M.A. Rojas-Medar, Exponential stability for magneto-micropolar fluids, Nonlinear Anal. 143 (2016), 211–223.
[28] Z. Tan, W. Wu, and J. Zhou, Global existence and decay estimate of solutions to magneto-micropolar fluid equations, J. Differ. Equ. 266 (2019), no. 7, 4137–4169.
[29] F. Wu, A refined regularity criteria of weak solutions to the magneto-micropolar fluid equations, J. Evol. Equ. 21 (2021), no. 1, 725—734.
[30] X. Xu, Z. Ye and Z. Zhang, Remark on an improved regularity criterion for the 3D MHD equations, Appl. Math. Lett. 42 (2015), 41–46.
[31] B.Q. Yuan, On regularity criteria of weak solutions to the micropolar fluid equations in Lorentz space, Proc. Amer. Math. Soc. 138 (2010), no. 6, 2010–2025.
[32] B. Yuan and X. Li, Regularity of weak solutions to the 3D magneto-micropolar equations in Besov spaces, Acta Appl. Math. 163 (2019), no. 1, 207–223.
[33] 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.
[34] Z. Zhang, Remarks on the global regularity criteria for the 3D MHD equations via two components, Z. Angew Math. 66 (2015), no. 3, 977–987.
[35] Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Int. J. Nonlinear Mech. 41 (2006), no. 10, 1174–1180.
International Journal of Nonlinear Analysis and Applications
Volume 15, Issue 12
December 2024
Pages 1-10
  • Receive Date: 29 September 2022
  • Accept Date: 05 October 2023