Archive for Rational Mechanics and Analysis

, Volume 213, Issue 2, pp 689–703 | Cite as

Periodic Motions of Stokes and Navier–Stokes Flows Around a Rotating Obstacle

  • Thieu Huy NguyenEmail author


We prove the existence and uniqueness of periodic motions to Stokes and Navier–Stokes flows around a rotating obstacle \({D \subset \mathbb{R}^3}\) with the complement \({\Omega = \mathbb{R}^3 \backslash D}\) being an exterior domain. In our strategy, we show the C b -regularity in time for the mild solutions to linearized equations in the Lorentz space \({L^{3,\infty}(\Omega)}\) (known as weak-L 3 spaces) and prove a Massera-typed Theorem on the existence and uniqueness of periodic mild solutions to the linearized equations in weak-L 3 spaces. We then use the obtained results for such equations and the fixed point argument to prove such results for Navier–Stokes equations around a rotating obstacle. We also show the stability of such periodic solutions.


Periodic Solution Periodic Motion Mild Solution Lorentz Space Stokes Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-valued Laplace transform and Cauchy problems. In: Monographs in Mathematics, vol. 96. Birkhäuser Verlag, Basel (2001)Google Scholar
  2. 2.
    Bergh J., Löfström J.: Interpolation Spaces. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  3. 3.
    Borchers W., Miyakawa T.: On stability of exterior stationary Navier–Stokes flows. Acta Math. 174, 311–382 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Farwig R., Hishida T.: Stationary Navier–Stokes flows around a rotating obstacle. Funkc. Ekvac. 50, 371–403 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Galdi G.P., Sohr H.: Existence and uniqueness of time-periodic physically reasonable Navier–Stokes flows past a body. Arch. Ration. Mech. Anal. 172, 363–406 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Galdi G.P., Silvestre A.L.: Existence of time-periodic solutions to the Navier Stokes equations around a moving body. Pac. J. Math. 223, 251–267 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Galdi G.P., Silvestre A.L.: On the motion of a rigid body in a Navier–Stokes liquid under the action of a time-periodic force, Indiana Univ. Math. J. 58, 2805–2842 (2009)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Geissert M., Heck H., Hieber M.: L p-Theory of the Navier–Stokes flow in the exterior of a moving or rotating obstacle. J. Reine Angew. Math. 596, 45–62 (2006)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Heywood J.G.: The Navier–Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29, 639–681 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Hishida T., Shibata Y.: L pL q Estimate of the Stokes operator and Navier–Stokes flows in the exterior of a rotating obstacle. Arch. Ration. Mech. Anal. 193, 339–421 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Kang K., Miura H., Tsai T.-P.: Asymptotics of small exterior Navier–Stokes flows with non-decaying boundary data. Commun. Part. Differ. Equ. 37, 1717–1753 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Kato T.: Strong L p-solutions of Navier–Stokes equations in \({{\mathbb R}^n}\) with applications to weak solutions. Math. Z. 187, 471–480 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Komatsu, H.: A general interpolation theorem of Marcinkiewicz type, Tôhoku Math. J. (2) 33, 383–393 (1981)Google Scholar
  14. 14.
    Maremonti P., Padula M.: Existence, uniqueness, and attainability of periodic solutions of the Navier–Stokes equations in exterior domains. J. Math. Sci. (New York), 93, 719–746 (1999)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Massera J.: The existence of periodic solutions of systems of differential equations. Duke Math. J. 17, 457–475 (1950)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Nagel, R. (ed.) One-Parameter Semigroups of Positive Operators. In: Lecture Notes in Mathematics, vol. 1184. Springer, New York (1986)Google Scholar
  17. 17.
    Nguyen T.H.: Invariant manifolds of admissible classes for semi-linear evolution equations. J. Differ. Equ. 246, 1820–1844 (2009)CrossRefzbMATHGoogle Scholar
  18. 18.
    Nguyen T.H.: Admissibly inertial manifolds for a class of semi-linear evolution equations. J. Differ. Equ. 254, 2638–2660 (2013)ADSCrossRefzbMATHGoogle Scholar
  19. 19.
    Prodi G.: Qualche risultato riguardo alle equazioni di Navier–Stokes nel caso bidimensionale. Rend. Sem. Mat. Univ. Padova 30, 1–15 (1960)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Prouse G.: Soluzioni periodiche dellequazione di Navier–Stokes. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 35, 443–447 (1963)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Salvi, R.: On the existence of periodic weak solutions on the Navier–Stokes equations in exterior regions with periodically moving boundaries. In: Sequeira, A. (eds.) Navier–Stokes Equations and Related Nonlinear Problems. Plenum, New York, pp. 63–73, (1995)Google Scholar
  22. 22.
    Serrin J.: A note on the existence of periodic solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 3, 120–122 (1959)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Taniuchi Y.: On the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domains. Math. Z. 261, 597–615 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam, (1978)Google Scholar
  25. 25.
    Van Baalen, G., Wittwer, P.: Time periodic solutions of the Navier–Stokes equations with nonzero constant boundary conditions at infinity. SIAM J. Math. Anal. 43, 1787–1809 (2011)Google Scholar
  26. 26.
    Yamazaki M.: The Navier–Stokes equations in the weak-L n space with time-dependent external force. Math. Ann. 317, 635–675 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Yoshizawa, T.: Stability theory and the existence of periodic solutions and almost periodic solutions. In: Applied Mathematical Sciences, vol. 14. Springer, New York (1975)Google Scholar
  28. 28.
    Yudovich V.: Periodic motions of a viscous incompressible fluid. Sov. Math. Dokl. 1, 168–172 (1960)zbMATHGoogle Scholar
  29. 29.
    Zubelevich O.: A note on theorem of Massera. Regul. Chaotic Dyn. 11, 475–481 (2006)ADSCrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.School of Applied Mathematics and InformaticsHanoi University of Science and TechnologyHanoiVietnam
  2. 2.Arbeitsgruppe Angewandte Analysis, Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany

Personalised recommendations