Applied Mathematics & Optimization

, Volume 79, Issue 1, pp 1–40 | Cite as

Gibbsian Dynamics and Ergodicity of Stochastic Micropolar Fluid System

  • Kazuo YamazakiEmail author


The theory of micropolar fluids emphasizes the micro-structure of fluids by coupling the Navier–Stokes equations with micro-rotational velocity, and is widely viewed to be well fit, better than the Navier–Stokes equations, to describe fluids consisting of bar-like elements such as liquid crystals made up of dumbbell molecules or animal blood. Following the work of Weinan et al. (Commun Math Phys 224:83–106, 2001), we prove the existence of a unique stationary measure for the stochastic micropolar fluid system with periodic boundary condition, forced by only the determining modes of the noise and therefore a type of finite-dimensionality of micropolar fluid flow. The novelty of the manuscript is a series of energy estimates that is reminiscent from analysis in the deterministic case.


Determining modes Ergodicity Micropolar fluid Navier–Stokes equations Stationary measure 

Mathematics Subject Classification

35Q35 37L55 60H15 


  1. 1.
    Ahmadi, G., Shahinpoor, M.: Universal stability of magneto-micropolar fluid motions. Int. J. Eng. Sci. 12, 657–663 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Batchelor, G.K.: The Theory of Homogeneous Turbulence. Cambridge University Press, Cambridge (1982)zbMATHGoogle Scholar
  3. 3.
    Bricmont, J., Kupiainen, A., Lefevere, R.: Ergodicity of the 2D Navier-Stokes equations with random forcing. Commun. Math. Phys. 224, 65–81 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Capiński, M., Peszat, S.: Local existence and uniqueness of strong solutions to 3-D stochastic Navier-Stokes equations. NoDEA Nonlinear Differ. Equ. Appl. 4, 185–200 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, Q., Miao, C.: Global well-posedness for the micropolar fluid system in critical Besov spaces. J. Differ. Equ. 252, 2698–2724 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Constantin, P., Foias, C.: Navier-Stokes Equations. Chicago Lectures in Mathematics . University of Chicago Press, Chicago (1988)Google Scholar
  7. 7.
    Da Prato, G., Debussche, A.: Ergodicity for the 3D Navier-Stokes equations. J. Math. Pures Appl. 82, 877–947 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Da Prato, G., Debussche, A., Temam, R.: Stochastic Burgers’ equation. NoDEA Nonlinear Differ. Equ. Appl. 1, 389–402 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  10. 10.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (2014)CrossRefzbMATHGoogle Scholar
  11. 11.
    Dong, B.-Q., Chen, Z.-M.: Regularity criteria of weak solutions to the three-dimensional micropolar flows. J. Math. Phys. 50, 103525 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dong, B.-Q., Chen, Z.-M.: Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows. Discret. Contin. Dyn. Syst. 23(3), 765–784 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dong, B.-Q., Zhang, Z.: Global regularity of the 2D micropolar fluid flows with zero angular viscosity. J. Differ. Equ. 249, 200–213 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    E, W., Liu, D.: Gibbsian dynamics and invariant measures for stochastic dissipative PDEs. J. Stat. Phys. 108, 1125–1156 (2002)Google Scholar
  15. 15.
    E, W., Mattingly, J.C., Sinai, Y.: Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Commun. Math. Phys. 224, 83–106 (2001)Google Scholar
  16. 16.
    Eringen, A.C.: Simple microfluids. Int. J. Eng. Sci. 2, 205–217 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Eringen, A.C.: Theory of micropolar fluids. J. Math. Mech. 16, 1–18 (1966)MathSciNetGoogle Scholar
  18. 18.
    Ferrario, B.: The Bénard problem with random perturbations: dissipativity and invariant measures. NoDEA Nonlinear Differ. Equ. Appl. 4, 101–121 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ferrario, B.: Ergodic results for stochastic navier-stokes equation. Stoch. Rep. 60, 271–288 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    B. Ferrario, Stochastic Navier-Stokes equations: analysis of the noise to have a unique invariant measure, Annali di Matematica pura ed applicata, CLXXVII, 331–347 (1999)Google Scholar
  21. 21.
    Flandoli, F.: Dissipativity and invariant measures for stochastic Navier-Stokes equations. NoDEA Nonlinear Differ. Equ. Appl. 1, 403–423 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Flandoli, F., Maslowski, B.: Ergodicity of the 2-D Navier-Stokes equation under random perturbations. Commun. Math. Phys. 171, 119–141 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Foias, C., Manley, O., Rosa, R., Temam, R.: Navier-Stokes Equations and Turbulence. Cambridge University Press, Cambridge, United Kingdom (2004)zbMATHGoogle Scholar
  24. 24.
    Foias, C., Prodi, G.: Sur le comportement global des solutions non-stationnaires des e’quations de Navier-Stokes en dimension 2. Rend. Semin. Mat. Univ. Padova 39, 1–34 (1967)zbMATHGoogle Scholar
  25. 25.
    Galdi, G.P., Rionero, S.: A note on the existence and uniqueness of solutions of the micropolar fluid equations. Int. J. Eng. Sci. 15, 105–108 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hairer, M., Mattingly, J.C.: Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann. Math. 164, 993–1032 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hmidi, T., Keraani, S., Rousset, R.: Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation. J. Differ. Equ. 249, 2147–2174 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Krylov, N., Bogoliubov, N.: La théorie générale de la mesure dans son application a l’étude des systémes de la mécanique nonlinéaire. Ann. Math. 38, 65–113 (1937)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Lee, J., Wu, M.-Y.: Ergodicity for the dissipative Boussinesq equations with random forcing. J. Stat. Phys. 117, 929–973 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lukaszewicz, G.: Micropolar Fluids, Theory and Applications. Birkh\(\ddot{a}\)user, Boston (1999)Google Scholar
  31. 31.
    Mattingly, J. C.: The Stochastic Navier-Stokes Equation, Energy Estimates and Phase Space Contraction, Ph.D. Thesis, Princeton University, Princeton (1998)Google Scholar
  32. 32.
    Mattingly, J.C.: Ergodicity of 2D Navier-Stokes equations with random forcing and large viscosity. Commun. Math. Phys. 206, 273–288 (1999)CrossRefzbMATHGoogle Scholar
  33. 33.
    Mattingly, J.C.: Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics. Commun. Math. Phys. 230, 421–462 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ortega-Torres, E.E., Rojas-Medar, M.A.: Magneto-micropolar fluid motion: global existence of strong solutions. Abstr. Appl. Anal. 4, 109–125 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Rojas-Medar, M.A.: Magneto-micropolar fluid motion: existence and uniqueness of strong solutions. Math. Nachr. 188, 301–319 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Wu, M.-Y.: Stochastic Boussinesq equations and the infinite dimensional Malliavin calculus, Ph.D. Thesis, Princeton University, Princeton (2006)Google Scholar
  37. 37.
    Xue, L.: Wellposedness and zero microrotation viscosity limit of the 2D micropolar fluid equations. Math. Methods Appl. Sci. 34, 1760–1777 (2011)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Yamazaki, K.: 3-D stochastic micropolar and magneto-micropolar fluid systems with non-Lipschitz multiplicative noise. Commun. Stoch. Anal. 8, 413–437 (2014)MathSciNetGoogle Scholar
  39. 39.
    Yamazaki, K.: Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity. Discrete Contin. Dyn. Syst. 35, 2193–2207 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Yamazaki, K.: Recent Developments on the Micropolar and Magneto-micropolar Fluid Systems: Deterministic and Stochastic Perspectives, Stochastic Equations for Complex Systems: Mathematical Engineering, pp. 85–103. Springer, New York (2015)Google Scholar
  41. 41.
    Yamazaki, K.: Ergodicity of the two-dimensional magnetic Benard problem. Electron. J. Differ. Equ. 2016, 1–24 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Yamazaki, K.: Exponential convergence of the stochastic micropolar and magneto-micropolar fluid systems. Commun. Stoch. Anal. 10, 271–295 (2016)MathSciNetGoogle Scholar
  43. 43.
    Yamazaki, K.: Remarks on the Large Deviation Principle for the Micropolar, Magneto-Micropolar Fluid Systems, submittedGoogle Scholar
  44. 44.
    Yuan, B.: On regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space. Proc. Am. Math. Soc. 138(6), 2025–2036 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of RochesterRochesterUSA

Personalised recommendations