Abstract
This paper is devoted to investigating regularity criteria for the 3D micropolar fluid equations in terms of pressure in weak Lebesgue space. More precisely, we mainly proved that the weak solution is regular on (0, T] provided that either the norm \(\left\| \pi \right\| _{L^{\alpha ,\infty }(0,T;L^{\beta ,\infty }(\mathbb {R}^{3}))}\) with \(\frac{2}{\alpha }+ \frac{3}{\beta }=2\) and \(\frac{3}{2}<\beta <\infty \) or \(\left\| \nabla \pi \right\| _{L^{\alpha ,\infty }(0,T;L^{\beta ,\infty }(\mathbb {R} ^{3}))}\) with \(\frac{2}{\alpha }+\frac{3}{\beta }=3\) and \(1<\beta <\infty \) is sufficiently small.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beirão da Veiga, H.: A sufficient condition on the pressure for the regularity of weak solutions to the Navier–Stokes equations. J. Math. Fluid Mech. 2, 99–106 (2000)
Beirão da Veiga, H.: Concerning the regularity of the solutions to the Navier–Stokes equations via the truncation method. In: II. Équations aux dérivées partielles et applications, pp. 127-138, Gauthier-Villars, Éd. Sci. Méd. Elsevier, Paris (1998)
Bergh, J., Löfström, J.: Interpolation Spaces. Springer, New York (1976)
Berselli, L., Galdi, G.: Regularity criteria involving the pressure for the weak solutions of the Navier–Stokes equations. Proc. Am. Math. Soc. 130, 3585–3595 (2002)
Bosia, S., Pata, V., Robinson, J.: A weak-\(L^{p}\) Prodi-Serrin type regularity criterion for the Navier–Stokes equations. J. Math. Fluid Mech. 16, 721–725 (2014)
Chen, J., Chen, Z.-M., Dong, B.-Q.: Uniform attractors of non-homogeneous micropolar fluid flows in non-smooth domains. Nonlinearity 20, 1619–1635 (2007)
Chen, Q., Miao, C.: Global well-posedness for the micropolar fluid system in critical Besov spaces. J. Differ. Equ. 252, 2698–2724 (2012)
Chen, Z.-M., Price, W.: Decay estimates of linearized micropolar fluid flows in \(\mathbb{R} ^{3}\) space with applications to \(L^{3}\) -strong solutions. Internat. J. Engrg. Sci. 44, 859–873 (2006)
Dong, B.-Q., Chen, Z.-M.: Regularity criteria of weak solutions to the three-dimensional micropolar flows. J. Math. Phys. 50, 1–13 (2009)
Dong, B.-Q., Zhang, W.: On the regularity criterion for the 3D micropolar fluid flows in Besov spaces. Nonlinear Anal. Theory Methods Appl. 73, 2334–2341 (2010)
Dong, B.-Q., Jia, Y., Chen, Z.-M.: Pressure regularity criteria of the three-dimensional micropolar fluid flows. Math. Meth. Appl. Sci. 34, 595–606 (2011)
Eringen, A.C.: Theory of micropolar fluids. J. Math. Mech. 16, 1–18 (1966)
Gala, S.: On regularity criteria for the three-dimensional micropolar fluid equations in the critical Morrey–Campanato space. Nonlinear Anal. Real World Appl. 12, 2142–2150 (2011)
Gala, S., Ragusa, M.A.: A regularity criterion for 3D micropolar fluid flows in terms of one partial derivative of the velocity. Ann. Polon. Math. 116, 217–228 (2016)
Gala, S., Yan, J.: Two regularity criteria via the logarithmic of the weak solutions to the micropolar fluid equations. J. Partial Differ. Equ. 25, 32–40 (2012)
Gala, S.: A remark on the logarithmically improved regularity criterion for the micropolar fluid equations in terms of the pressure. Math. Meth. Appl. Sci. 34, 1945–1953 (2011)
Galdi, G., Rionero, S.: A note on the existence and uniqueness of solutions of micropolar fluid equations. Int. J. Engrg. Sci. 14, 105–108 (1977)
Grafakos, L.: Classical Fourier analysis, 2nd edn. Springer (2008)
Ji, X., Wang, Y., Wei, W.: New regularity criteria based on pressure or gradient of velocity in Lorentz spaces for the 3D Navier–Stokes equations. J. Math. Fluid Mech. 22(1), 13 (2020)
Jia, Y., Zhang, W., Dong, B.: Remarks on the regularity criterion of the 3D micropolar fluid flows in terms of the pressure. Appl. Math. Lett. 24, 199–203 (2011)
Jia, Y., Zhang, W., Dong, B.: Logarithmical regularity criteria of the three-dimensional micropolar fluid equations in terms of the pressure. Abstr. Appl. Anal. (2012). https://doi.org/10.1155/2012/395420
Kozono, H., Yamazaki, M.: Exterior problem from the stationary Navier-Stokes equations in the Lorentz space. Math. Ann. 310, 279–305 (1998)
Lukaszewicz, G.: Micropolar fluids. Theory and Applications. Modeling and Simulation in Science, Engineering and Technology. Birkhauser Boston, Inc, Boston, MA (1999)
Loayza, M., Rojas-Medar, M.A.: A weak-\(L^{p}\) Prodi-Serrin type regularity criterion for the micropolar fluid equations. J. Math. Phys. 57, 021512 (2016)
Malý, J.: Advanced theory of differentiation-Lorentz
Pineau, B., Yu, X.: A new Prodi-Serrin type regularity criterion in velocity directions. J. Math. Fluid Mech. 20, 1737–1744 (2018)
Pineau, B., Yu, X.: On Prodi-Serrin type conditions for the 3D Navier–Stokes equations. Nonlinear Anal. 190, 111612 (2020)
Rojas-Medar, M.: Magnato-microplar fluid motion: existence and uniqueness of strong solution. Math. Nachr. 188, 301–319 (1997)
Suzuki, T.: Regularity criteria of weak solutions in terms of the pressure in Lorentz spaces to the Navier–Stokes equations. J. Math. Fluid Mech. 14, 653–660 (2012)
Suzuki, T.: A remark on the regularity of weak solutions to the Navier–Stokes equations in terms of the pressure in Lorentz spaces. Nonlinear Anal. Theory Methods Appl. 75, 3849–3853 (2012)
Triebel, H.: Theory of Function Spaces. Birkhäuser Verlag, Basel, Boston (1983)
Yamaguchi, N.: Existence of global strong solution to the micropolar fluid equations. Math. Methods Appl. Sci. 28, 1507–1526 (2005)
Yuan, B.: On the regularity criteria of weak solutions to the micropolar fluid equations in Lorentz space. Proc. Am. Math. Soc. 138, 2025–2036 (2010)
Wang, Y., Zhao, H.: Logarithmically improved blow up criterion for smooths solution to the 3D micropolar fluid equations. J. Appl. Math. 10, 1–13 (2012)
Zhou, Y.: Regularity criteria in terms of pressure for the 3-D Navier–Stokes equations in a generic domain. Math. Annalen 328, 173–192 (2004)
Zhou, Y.: On the regularity criteria in terms of pressure for the Navier–Stokes equations in \(\mathbb{R} ^{3}\). Proc. Am. Math. Soc. 134, 149–156 (2006)
Zhou, Y.: On a regularity criterion in terms of the gradient of pressure for the Navier–Stokes equations in \(\mathbb{R} ^{3}\). Z. Angew. Math. Phys. 57, 384–392 (2006)
Zhou, Y.: Regularity criteria for the 3D MHD equations in terms of the pressure. Internat. J. Non-Linear Mech. 41, 1174–1180 (2006)
Acknowledgements
The authors wish to express their thanks to the referee for his/her very careful reading of the paper, giving valuable comments and helpful suggestions. M. Benslimane extends his appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia). The fourth author likes to thank Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Vietnam, for the opportunity to work in it. This paper has been supported by Faculty of Fundamental Science, Industrial University of Ho Chi Minh City and P.R.I.N.
Author information
Authors and Affiliations
Contributions
The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final paper.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Omrane, I.B., Slimane, M.B., Gala, S. et al. A weak-\(L^{p}\) Prodi–Serrin type regularity criterion for the micropolar fluid equations in terms of the pressure. Ricerche mat (2023). https://doi.org/10.1007/s11587-023-00829-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11587-023-00829-2