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Vanishing Viscosity Limit for the 3D Incompressible Micropolar Equations in a Bounded Domain

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Abstract

In this paper, we investigate the vanishing viscosity limit of the 3D incompressible micropolar equations in bounded domains with boundary conditions. It is shown that there exist global weak solutions of the micropolar equations in a general bounded smooth domain. In particular, we establish the uniform estimate of the strong solutions for when the boundary is flat. Furthermore, we obtain the rate of convergence of viscosity solutions to the inviscid solutions as the viscosities tend to zero (i.e., (ε,χ,γ,κ) → 0).

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References

  1. Aero E L, Bulganin A N, Kuvshinski E V. Asymmetric hydrodynamics. Prikl Mat Mech, 1965, 29(2): 297–308

    Google Scholar 

  2. Lukaszewicz G. Micropolar Fluid: Theory and Applications, Modeling and Simulation in Science, Engineering and Thchnology. Boston: Birkhäuser, 1999

    Book  Google Scholar 

  3. Li F C, Zhang Z P. Zero Kinematic viscosity-magnetic diffusion limit of the incompressible viscous magnetohydrodynamic equations with Navier boundary conditions. Acta Math Sci, 2021, 41B(5): 1503–1536

    Article  MATH  Google Scholar 

  4. Li H M, Xiao Y L. Local well-posedness of strong solutions for the nonhomogeneous MHD equations with a slip boundary conditions. Acta Math Sci, 2020, 40B(2): 442–456

    Article  MATH  Google Scholar 

  5. Nowakowski B. Global existence of strong silutions to micropolar equations in cylindrical domians. Math Methods Appl Sci, 2015, 38(2): 311–329

    Article  MATH  Google Scholar 

  6. Xiao Y L, Xin Z P. On the inviscid limit of the 3D Navier-Stokes equations with generalized Navier-Slip boundary conditions. Commun Math Stat, 2013, 1(3): 259–279

    Article  MATH  Google Scholar 

  7. Xiao Y L, Xin Z P. On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition. Comm Pure Appl Math, 2007, 60(7): 1027–1055

    Article  MATH  Google Scholar 

  8. Eringen A C. Theory of micropolar fluids. J Math Mech, 1966, 16: 1–18

    Google Scholar 

  9. Ladyzhenskaya O A. The Mathematical Theory of Viscous Incompressible Fluids. New York: Gordon and Breach, 1969

    MATH  Google Scholar 

  10. Chen Q L, Miao C X. Global well-posedness for the micropolar fluid system in the critical Besov spaces. J Differ Equ, 2012, 252(3): 2698–2724

    Article  MATH  Google Scholar 

  11. Dong B Q, Chen Z M. Regularity criteria of weak solutions to the three-dimensional micropolar flows. J Math Phys, 2009, 50(10): 103525

    Article  MATH  Google Scholar 

  12. Dong B Q, Li J N, Wu J H. Global well-posedness and large-time decay for the 2D micropolar equations. J Differ Equ, 2017, 262(6): 3488–3523

    Article  MATH  Google Scholar 

  13. Galdi G P, Rionero S. A note on the existence and uniqueness of solutions of micropolar fluid equations. Int J Engrg Sci, 1977, 15(2): 105–108

    Article  MATH  Google Scholar 

  14. Jiu Q S, Liu J T, Wu J H, et al. On the initial- and boundary-value problem for 2D micropolar equations with only angular velocity dissipation. Z Angew Math Phys, 2017, 68(5): Art 107

    Article  MATH  Google Scholar 

  15. Liu J T, Wang S. Initial-boundary value problem for 2D micropolar equations without angular viscosity. Commun Math Sci, 2018, 16(8): 2147–2165

    Article  MATH  Google Scholar 

  16. Ye Z. Global existence of strong solution to the 3D micropolar equations with a damping term. Appl Math Lett, 2018, 83: 188–193

    Article  MATH  Google Scholar 

  17. Yamaguchi N. Existence of global strong solution to the micropolar fluid system in a bounded domain. Math Methods Appl Sci, 2005, 28(13): 1507–1526

    Article  MATH  Google Scholar 

  18. Ferreira L C F, Villamizar-Roa E J. On the existence and stability of solutions for the micropolar fluids in exterior domains. Math Meth Appl Sci, 2007, 30(10): 1185–1208

    Article  MATH  Google Scholar 

  19. Yang Y Y, Zhu M X. The zero limit of angular viscosity for the two-Dimensional micropolar fluid equations. Appl Math Lett, 2016, 57: 32–37

    Article  MATH  Google Scholar 

  20. Chen M T, Xu X Y, Zhang J W. The zero limits of angular and micro-rotational viscosities for the two-dimensional micropolar fluid equations with boundary effect. Z Angew Math Phys, 2014, 65(4): 687–710

    Article  MATH  Google Scholar 

  21. Beirão da Veiga H, Crispo F. The 3-D inviscid limits result under slip boundary conditions. A negative answer. J Math Fluid Mech, 2012, 14(1): 55–59

    Article  MATH  Google Scholar 

  22. Berselli L C. Some results on the Navier-Stokes equations with Navier boundary conditions. Riv Math Univ Parma, 2010, 1(1): 1–75

    MATH  Google Scholar 

  23. Iftimie D, Planas G. Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions. Nonlinearity, 2006, 19(4): 899–918

    Article  MATH  Google Scholar 

  24. Kato T. Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary. Math Sci Res Inst Publ, 1984, 2: 85–98

    MATH  Google Scholar 

  25. Masmoudi N. Remarks about the inviscid limit of the Navier-Stokes system. Comm Math Phys, 2007, 270(3): 777–788

    Article  MATH  Google Scholar 

  26. Bourguignon J P, Brezis H. Remarks on the Euler equation. J Funct Anal, 1974, 15: 341–363

    Article  MATH  Google Scholar 

  27. Yoshida Z, Giga Y. Remarks on spectra of operator rot. Math Z, 1990, 204(2): 235–245

    Article  MATH  Google Scholar 

  28. Ortega-Torres E E, Rojas-Medar M A. Magneto-micropolar fluid motion: global existence of strong solutions. Abstr Appl Anal, 1999, 4(2): 109–125

    Article  MATH  Google Scholar 

  29. Rojas-Medar M A. Magneto-micropolar fluid motion: Existence and uniqueness of strong solution. Math Nachr, 1997, 188: 301–319

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China

    Yangyang Chu  (储洋) & Yuelong Xiao  (肖跃龙)

Authors
  1. Yangyang Chu  (储洋)
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  2. Yuelong Xiao  (肖跃龙)
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Corresponding author

Correspondence to Yuelong Xiao  (肖跃龙).

Additional information

This work was supported by the NSFC (11871412) and the Postgraduate Scientific Research Innovation Project of Xiangtan University (XDCX2020B088).

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Chu, Y., Xiao, Y. Vanishing Viscosity Limit for the 3D Incompressible Micropolar Equations in a Bounded Domain. Acta Math Sci 43, 959–974 (2023). https://doi.org/10.1007/s10473-023-0224-2

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  • DOI: https://doi.org/10.1007/s10473-023-0224-2

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