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Large Time Behavior of Spherically Symmetrical Micropolar Fluid on Unbounded Domain

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Abstract

This paper is concerned with the large-time behavior of solutions to the initial boundary value problem with large initial data for the compressible micropolar fluid model on an unbounded domain exterior to the unit ball in \({\mathbb {R}}^3\). Based on the energy-estimate method, we first show that the absolute temperature is uniformly bounded in time and then obtain the asymptotic stability of global solution.

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References

  1. Chen, M.T.: Global strong solutions for the viscous, micropolar, compressible flow. J. Partial Differ. Equ. 24, 158–164 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cui, H., Yin, H.: Stationary solutions to the one-dimensional micropolar fluid model in a half line: existence, stability and convergence rate. J. Math. Anal. Appl. 449, 464–489 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dražić, I.: 3-D flow of a compressible viscous micropolar fluid with cylindrical symmetry: a global existence theorem. Math. Method Appl. Sci. 40, 4785–4801 (2017)

    MathSciNet  MATH  Google Scholar 

  4. Dražić, I., Mujaković, N.: 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: a local existence theorem. Bound. Value Probl. 2012, 69 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dražić, I., Mujaković, N.: 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: uniqueness of a generalized solution. Bound. Value Probl. 2014, 226 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dražić, I., Mujaković, N.: 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: a global existence theorem. Bound. Value Probl. 2015, 98 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dražić, I., Mujaković, N.: 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: large time behavior of the solution. J. Math. Anal. Appl. 431, 545–568 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dražić, I., Mujaković, N., Črnjarić-Žic, N.: Three-dimensional compressible viscous micropolar fluid with cylindrical symmetry: derivation of the model and a numerical solution. Math. Comput. Simul. 140, 107–124 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dražić, I., Simčić, L., Mujaković, N.: Three-dimensional compressible viscous micropolar fluid with cylindrical symmetry: uniqueness of a generalized solution. Math. Method Appl. Sci. 40, 2686–2701 (2017)

    Article  MATH  Google Scholar 

  10. Duan, R.: Global solutions for a one-dimensional compressible micropolar fluid model with zero heat conductivity. J. Math. Anal. Appl. 463, 477–495 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  12. Eringen, A.C.: Microcontinuum Field Theories: I. Foundations and Solids. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  13. Evans, L.C.: Partial Differential Equations. Graduate studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence (2010)

    Google Scholar 

  14. Huang, L., Dražić, I.: Large-time behavior of solutions to the 3-D flow of a compressible viscous micropolar fluid with cylindrical symmetry. Math. Method Appl. Sci. 41, 7888–7905 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huang, L., Dražić, I.: Exponential stability for the compressible micropolar fluid with cylinder symmetry in $R^3$. J. Math. Phys. 60, 021507 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Huang, L., Kong, C.: Global behavior for compressible viscous micropolar fluid with spherical symmetry. J. Math. Anal. Appl. 443, 1158–1178 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Huang, L., Lian, R.: Exponential stability of spherically symmetric solutions for compressible viscous micropolar fluid. J. Math. Phys. 56, 071503 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Huang, L., Nie, D.: Exponential stability for a one-dimensional compressible viscous micropolar fluid. Math. Method Appl. Sci. 38, 5197–5206 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Huang, F., Qin, X.: Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation. J. Differ. Equ. 246, 4077–4096 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Huang, L., Yang, X., Lu, Y., Wang, T.: Global attractors for a nonlinear one-dimensional compressible viscous micropolar fluid model. Z. Angew. Math. Phys. 70, 1–20 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jiang, S.: Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain. Commun. Math. Phys. 178, 339–374 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jiang, S.: Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains. Commun. Math. Phys. 200, 181–193 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kawashima, S., Nishibata, S., Zhu, P.: Asymptotic stability of the stationary solution to the compressible Navier–Stokes equations in the half space. Commun. Math. Phys. 240, 483–500 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, J., Liang, Z.: Some uniform estimates and large-time behavior for one-dimensional compressible Navier–Stokes system in unbounded domains with large data. Arch. Ration. Mech. Anal. 220, 1195–1208 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Liang, Z.: Large-time behavior for spherically symmetric flow of viscous polytropic gas in exterior unbounded domain with large initial data. http://arxiv.org/abs/1405.0569

  26. Liao, Y., Zhao, H.: Global existence and large-time behavior of solutions to the Cauchy problem of one-dimensional viscous radiative and reactive gas. J. Differ. Equ. 265, 2076–2120 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liao, Y., Wang, T., Zhao, H.: Global spherical symmetric flows for a viscous radiative and reactive gas in an exterior domain. J. Differ. Equ. 266, 6459–6506 (2019)

    Article  MATH  Google Scholar 

  28. Lukaszewicz, G.: Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology. Birkhauser, Baston (1999)

    MATH  Google Scholar 

  29. Mujaković, N.: One-dimensional flow of a compressible viscous micropolar fluid: a local existence theorem. Glasnik Matematicki 33, 71–91 (1998)

    MathSciNet  MATH  Google Scholar 

  30. Mujaković, N.: One-dimensional flow of a compressible viscous micropolar fluid: a global existence theorem. Glasnik Matematicki 33, 199–208 (1998)

    MathSciNet  MATH  Google Scholar 

  31. Mujaković, N.: Global in time estimates for one-dimensional compressible viscous micropolar fluid model. Glasnik Matematicki 40, 103–120 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mujaković, N.: One-dimensional flow of a compressible viscous micropolar fluid: the Cauchy problem. Math. Commun. 10, 1–14 (2005)

    MathSciNet  MATH  Google Scholar 

  33. Mujaković, N.: Uniqueness of a solution of the Cauchy problem for one-dimensional compressible viscous micropolar fluid model. Appl. Math. E-Notes 6, 113–118 (2006)

    MathSciNet  MATH  Google Scholar 

  34. Mujaković, N.: Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: a local existence theorem. Ann. Univ. Ferrara 53, 361–379 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Mujaković, N.: Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid: a global existence theorem. Math. Inequal. Appl. 12, 651–662 (2009)

    MathSciNet  MATH  Google Scholar 

  36. Mujaković, N.: 1-D compressible viscous micropolar fluid model with non-homogeneous boundary conditons for temperature: a local existence theorem. Nonlinear Anal. Real World Appl. 13, 1844–1853 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mujaković, N.: The existence of a global solution for one dimensional compressible viscous micropolar fluid with non-homogeneous boundary conditions for temperature. Nonlinear Anal. Real World Appl. 19, 19–30 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Mujaković, N., Črnjarić-Z̆ic, N.: Convergent finite difference scheme for 1D flow of compressible micropolar fluid. Int. J. Numer. Anal. Model. 12, 94–124 (2015)

    MathSciNet  MATH  Google Scholar 

  39. Matsumura, A., Mei, M.: Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary. Arch. Ration. Mech. Anal. 146, 1–22 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  40. Matsumura, A., Nishihara, K.: Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas. Commun. Math. Phys. 222, 449–474 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  41. Nakamura, T., Nishibata, S.: Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas. J. Hyperbolic Differ. Equ. 8, 651–670 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. Nakamura, T., Nishibata, S., Yuge, T.: Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line. J. Differ. Equ. 241, 94–111 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  43. Qin, Y., Wang, T., Hu, G.: The Cauchy problem for a 1D compressible viscous micropolar fluid model: Analysis of the stabilization and the regularity. Nonlinear Anal. Real World Appl. 12, 1010–1029 (2013)

    MathSciNet  MATH  Google Scholar 

  44. Wan, L., Wang, T.: Large-time behavior of solutions to the equations of a viscous heat-conducting flow with shear viscosity in unbounded domains. J. Math. Anal. Appl. 436, 366–381 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhang, L.: Spherically symmetric flow for a viscous radiative and reactive gas in a field of potential forces with large initial data. Math. Method Appl. Sci. 42, 2785–2824 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Lan Huang was supported by the NSFC (No. 11871212 and No. 11501199) and the Natural Science Foundation of Henan Province (No. 20A110026). Yongjin Lu was partially supported by NSF (No: 1601127). Xin-Guang Yang was partially supported by the fund of Young Backbone Teacher in Henan Province (No. 2018GGJS039), Incubation Fund from Henan Normal University (No. 2020PL17) and Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003).

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Authors and Affiliations

  1. College of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou, 450011, People’s Republic of China

    Lan Huang & Zhiying Sun

  2. Department of Mathematics and Economics, Virginia State University, Petersburg, VA, 23806, USA

    Yongjin Lu

  3. Department of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, Henan, People’s Republic of China

    Xin-Guang Yang

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  1. Lan Huang
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Huang, L., Sun, Z., Lu, Y. et al. Large Time Behavior of Spherically Symmetrical Micropolar Fluid on Unbounded Domain. Appl Math Optim 84 (Suppl 2), 1607–1638 (2021). https://doi.org/10.1007/s00245-021-09806-3

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