Abstract
This paper is concerned with the large time decay rates of the two-dimensional (2D) micropolar equations with zero angular viscosity. Based on the generalized Fourier splitting methods and low frequency effect analysis, we firstly obtain the solutions decay as \(\Vert u\Vert _{L^2}+ \Vert \omega \Vert _{L^2}\le C(1+t)^{-\frac{1}{2}}\). Moreover, by exploring the new structure of the system, we obtain a new improved decay rates \(\Vert \omega \Vert _{L^2}+ \Vert \nabla u\Vert _{L^2}\le C(1+t)^{-1}\). Our methods here are also available to the time decay issue of the complex fluid flows with partial dissipation.
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References
Dong, B., Chen, Z.: On upper and lower bounds of higher order derivatives for solutions to the 2D micropolar fluid equations. J. Math. Anal. Appl. 334, 1386–1399 (2007)
Dong, B., Chen, Z.: Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows. Discrete Contin. Dyn. Syst. 23, 765–784 (2009)
Dong, B., Zhang, Z.: Global regularity of the 2D micropolar fluid flows with zero angular viscosity. J. Differ. Equ. 249, 200–213 (2010)
Dong, B., Li, J., Wu, J.: Global well-posedness and large-time decay for the 2D micropolar equaitons. J. Differ. Equ. 262, 3488–3523 (2017)
Eringen, A.C.: Theory of micropolar fluids. J. Math. Mech. 16, 1–18 (1966)
Galdi, G.P., Rionero, S.: A note on the existence and uniqueness of solutions of micropolar fluid equations. Int. J. Eng. Sci. 14, 105–108 (1977)
Jiu, Q., Liu, J., Wu, J., Yu, H.: On the initial- and boundary-value problem for 2D micropolar equations with only angular velocity dissipation. Z. Angew. Math. Phys. 68, 24 (2017)
Łukaszewicz, G.: Micropolar fluids. Theory and applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston (1999)
Lange, H.: The existence of instationary flows of incompressible micropolar fluids. Arch. Mech. 29, 741–744 (1977)
Resndiz, B., Rojas-Medar, M.: Existence of weak solution of micropolar fluid equations in a time dependent domain. Rev. Mat. Apl. 23, 27–46 (2002)
Schonbek, M.E.: \(L^{2}\) decay for weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 88, 209–222 (1985)
Teman, R.: The Navier–Stokes Equations. North-Holland, Amsterdam (1977)
Xue, L.: Well posedness and zero microrotation viscosity limit of the 2D micropolar fluid equations. Math. Methods Appl. Sci. 34, 1760–1777 (2011)
Zhao, C., Li, B.: Time decay rate of weak solutions to the generalized MHD equations in \(R_{2}\). Appl. Math. Comput. 292, 1–8 (2017)
Zhao, C., Liang, Y., Zhao, M.: Upper and lower bounds of time decay rate of solutions to a class of incompressible third grade fluid equations. Nonlinear Anal. Real World Appl. 15, 229–238 (2014)
Zhao, X., Samet, B., Zhou, Y.: Decay of solutions to a class of incompressible third grade fluid equations. J. Math. Anal. Appl. 484, 123678 (2020)
Zhou, Y.: A remark on the decay of solutions to the 3-D Navier–Stokes equations. Math. Methods Appl. Sci. 30(10), 1223–1229 (2007)
Zhou, Y.: Asymptotic behaviour of the solutions to the 2D dissipative quasi-geostrophic flows. Nonlinearity 21(9), 2061–2071 (2008)
Acknowledgements
The authors want to express their sincere thanks to the editors and the referees for their invaluable comments and suggestions which helped improve the paper greatly. Dong is partially supported by the National Natural Science Foundation of China (No. 11871346), the Natural Science Foundation of Guangdong Province (No. 2018A030313024), NSF of Shenzhen City(No. JCYJ20180305125554234) and Research Fund of Shenzhen University (No. 2017056). Jia was supported by the NNSFC grants No. 11801002 and the NSF of Anhui Province (No. 1808085MA01).
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Communicated by Syakila Ahmad.
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Guo, Y., Jia, Y. & Dong, BQ. Time Decay Rates of the Micropolar Equations with Zero Angular Viscosity. Bull. Malays. Math. Sci. Soc. 44, 3663–3675 (2021). https://doi.org/10.1007/s40840-021-01138-3
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DOI: https://doi.org/10.1007/s40840-021-01138-3