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Stability and large time decay for the three-dimensional magneto-micropolar equations with mixed partial viscosity

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Abstract

This paper focuses on the stability problem and decay estimates for two classes of three-dimensional (3D) magneto-micropolar equations with mixed partial viscosity. When \(\mu _3= \nu _3 = \gamma _1 = \gamma _2=0\) and \(\chi \Delta u \) replaced by \(\chi u\) in (1.2), by fully exploiting the structure of the system, and the method of bootstrapping argument, we prove the global stability of solutions to this system with initial data small in \(H^2({\mathbb {R}}^3)\). Furthermore, for these global solutions with initial data in \(H^s({\mathbb {R}}^3)\) \((s \ge 3)\) being large, we obtain their global \(H^s({\mathbb {R}}^3)\) \((s \ge 3)\) bound which is independent of time. In addition, we obtain the global existence of solutions for small initial data and the decay estimates of these solutions to 3D magneto-micropolar equations with mixed partial viscosity [namely \(\mu _3= \gamma _1 = \gamma _2=0\) and \(\chi \Delta u \) replaced by \(\chi u\) in (1.2)].

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References

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

    MathSciNet  Google Scholar 

  2. Ahmadi, G., Shahinpoor, M.: Universal stability of magneto-micropolar fluid motions. Int. J. Eng. Sci. 12, 657–663 (1974)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. Rojas-Medar, M.: Magneto-micropolar fluid motion: existence and uniqueness of strong solution. Math. Nachr. 188, 301–319 (1997)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  6. Rojas-Medar, M., Boldrini, J.: Magneto-micropolar fluid motion: existence of weak solutions. Rev. Mat. Complut. 11, 443–460 (1998)

    MathSciNet  MATH  Google Scholar 

  7. Yuan, b: Regularity of weak solutions to magneto-micropolar fluid equations. Acta Math. Sci. 30, 1469–1480 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Xu, F.: Regularity criterion of weak solution for the 3D magneto-micropolar fluid equations in Besov spaces. Commun. Nonlinear Sci. 17, 2426–2433 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Yuan, J.: Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations. Math. Methods Appl. Sci. 31, 1113–1130 (2008)

    MathSciNet  MATH  Google Scholar 

  10. Gala, S.: Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey–Campanato space. Nonlinear Differ. Equ. Appl. 17, 181–194 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Zhang, Z., Yao, Z., Wang, X.: A regularity criterion for the 3D magneto-micropolar fluid equations in Triebel–Lizorkin spaces. Nonlinear Anal. 74, 2220–2225 (2011)

    MathSciNet  MATH  Google Scholar 

  12. Wang, Y., Gu, L.: Global regularity of 3D magneto-micropolar fluid equations. Appl. Math. Lett. 99, 105980 (2020)

    MathSciNet  MATH  Google Scholar 

  13. Liu, Y.: Global regularity for the 2D magneto-micropolar system with partial and fractional dissipation. Math. Methods Appl. Sci. 43, 1–25 (2020)

    MathSciNet  Google Scholar 

  14. Dong, B., Zhang, Z.: Global regularity of the 2D micropolar fluid flows with zero angular viscosity. J. Differ. Equ. 249, 200–213 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Cao, C., Wu, J., Yuan, B.: The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion. SIAM J. Math. Anal. 46, 588–602 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Regmi, D., Wu, J.: Global regularity for the 2D magneto-micropolar equations with partial dissipation. J. Math. Study 49, 169–194 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Regmi, D.: The 2D magneto-micropolar equations with partial dissipation. Math. Methods Appl. Sci. 42, 4305–4317 (2019)

    MathSciNet  MATH  Google Scholar 

  18. Guo, Y., Shang, H.: Global well-posedness of two-dimensional magneto-micropolar equations with partial dissipation. Appl. Math. Comput. 313, 392–407 (2017)

    MathSciNet  MATH  Google Scholar 

  19. Wang, F., Wang, K.: Global existence of 3D MHD equations with mixed partial dissipation and magnetic diffusion. Nonlinear Anal. Real World Appl. 14, 526–535 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Wu, J., Zhu, Y.: Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium. Adv. Math. 377(21), 107466 (2021)

    MathSciNet  MATH  Google Scholar 

  21. Wang, Y., Wang, K.: Global well-posedness of 3D magneto-micropolar fluid equations with mixed partial viscosity. Nonlinear Anal. Real World Appl. 33, 348–362 (2017)

    MathSciNet  MATH  Google Scholar 

  22. Wang, Y., Li, W.: Global well-posedness of 3D magneto-micropolar fluid equations with mixed partial viscosity near an equilibrium. Z. Angew. Math. Phys. 72, 1–23 (2021)

    MathSciNet  MATH  Google Scholar 

  23. Shang, H., Zhai, Y.: Stability and large time decay for the three-dimensional anisotropic magnetohydrodynamic equations. Z. Angew. Math. Phys. 73, 1–22 (2022)

    MathSciNet  MATH  Google Scholar 

  24. Ai, C., Tan, Z.: Large time decay rates of the two-dimensional magneto-micropolar fluids with zero angular viscosity. Appl. Math. Lett 132, 108104 (2022). (8)

    MathSciNet  MATH  Google Scholar 

  25. Guterres, R., Melo, W., Nunes, J., Perusato, C.: On the large time decay of asymmetric flows in homogeneous Sobolev spaces. JMAA 471, 88–101 (2019)

    MathSciNet  MATH  Google Scholar 

  26. Cruz, F., Perusato, C., Rojas-Medar, M., Zingano, P.: Large time behavior for MHD micropolar fluids in \({\mathbb{R} }^n\). J. Differ. Equ. 312, 1–44 (2022)

    MATH  Google Scholar 

  27. Guo, Y., Jia, Y., Dong, B.: Existence and time decay for global small solution of 2D generalized magneto-micropolar equations. Acta Appl. Math. 174, 3 (2021)

    MathSciNet  MATH  Google Scholar 

  28. Guterres, R., Niche, César J., Perusato, C., Zingano, P.: Upper and lower \({\dot{H}}^m\) estimates for solutions to parabolic equations. J. Differ. Equ. 356 (2023)

  29. Guterres, R., Nunes, J., Perusato, C.: On the large time decay of global solutions for the micropolar dynamics in \(L^2({\mathbb{R} }^n)\). Nonlinear Anal. Real World Appl. 45, 789–798 (2019)

    MathSciNet  MATH  Google Scholar 

  30. Melo, W., Perusato, C., Guterres, R., Nunes, J.: Large time decay for the magnetohydrodynamics system in \({\dot{H}}({\mathbb{R} }^n)\). Acta Appl. Math. 68, 1–16 (2020)

    MATH  Google Scholar 

  31. Niche, C., Perusato, C., Sharp decay estimates and asymptotic behaviour for 3D magneto-micropolar fluids. Z. Angew. Math. Phys. 48 (2022)

  32. Kreiss, H., Hagstrom, T., Lorenz, J., Zingano, P.: Decay in time of incompressible flows. J. Math. Fluid Mech. 5, 231–244 (2003)

    MathSciNet  MATH  Google Scholar 

  33. Perusato, C., Melo, W., Guterres, R., Nunes, J.: Time asymptotic profiles to the magneto-micropolar system. Appl. Anal. 99, 2678–2691 (2020)

    MathSciNet  MATH  Google Scholar 

  34. Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)

    MathSciNet  MATH  Google Scholar 

  35. Schonbek, M.: \(L^2\) decay for weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 88, 209–222 (1985)

    MATH  Google Scholar 

  36. Schonbek, M.: Large time behaviour of solutions to the Navier–Stokes equations. Commun. Partial Differ. Equ. 11, 733–763 (1986)

    MathSciNet  MATH  Google Scholar 

  37. Brandolese, L.: Characterization of solutions to dissipative systems with sharp algebraic decay. SIAM J. Math. Anal. 48, 1616–1633 (2016)

    MathSciNet  MATH  Google Scholar 

  38. Niche, C., Schonbek, M.: Decay characterization of solutions to dissipative equations. J. Lond. Math. Soc. 91, 573–595 (2015)

    MathSciNet  MATH  Google Scholar 

  39. Li, M., Shang, H.: Large time decay of solutions for the 3D magneto-micropolar equations. Nonlinear Anal. Real World Appl. 44, 479–496 (2018)

    MathSciNet  MATH  Google Scholar 

  40. Shang, H., Gu, C.: Global regularity and decay estimates for 2D magneto-micropolar equations with partial dissipation. Z. Angew. Math. Phys. 70, 1–22 (2019)

    MathSciNet  MATH  Google Scholar 

  41. Shang, H., Gu, C.: Large time behavior for two-dimensional magneto-micropolar equations with only micro-rotational dissipation and magnetic diffusion. Appl. Math. Lett. 99, 1–7 (2019)

    MathSciNet  Google Scholar 

  42. M, Li.: Stability and time decay rates of the 2D magneto-micropolar equations with partial dissipation. Z. Angew. Math. Phys. 73(3), 100, 20 (2022)

    MathSciNet  MATH  Google Scholar 

  43. Niu, D., Shang, H., Global well-posedness and optimal time decay rates of solutions to the three-dimensional magneto-micropolar fluid equations, submitted for publication

  44. Yang, W., Jiu, Q., Wu, J.: The 3D incompressible Naver–Stokes equations with partial hyperdissipation. Math. Nachr. 292, 1823–1836 (2019)

    MathSciNet  MATH  Google Scholar 

  45. Lieb, E., Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society (2001)

    Google Scholar 

  46. Bahouri, H., Chemin, J., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations, Grundlrlehrender Mathematischen Wissenschaften, vol. 343. Springer, Berlin (2011)

    MATH  Google Scholar 

  47. Majda, A., Bertozzi, A.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  48. Miao, C., Yuan, B., Zhang, B.: Well-posedness of the Cauchy problem for the fractional power dissipative equations. Nonlinear Anal. 68, 461–484 (2008)

    MathSciNet  MATH  Google Scholar 

  49. Kenig, C., Ponce, G., Vega, L.: Well-posedness of the initial value problem for the Korteweg–de Vries equation. J. Am. Math. Soc. 4, 323–347 (1991)

    MathSciNet  MATH  Google Scholar 

  50. Ju, N.: Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space. Commun. Math. Phys. 251, 365–376 (2004)

    MathSciNet  MATH  Google Scholar 

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  1. Center for Nonlinear Studies, School of Mathematics, Northwest University, Xi’an, People’s Republic of China

    Ming Li

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Li, M. Stability and large time decay for the three-dimensional magneto-micropolar equations with mixed partial viscosity. Z. Angew. Math. Phys. 74, 110 (2023). https://doi.org/10.1007/s00033-023-01992-0

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