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On Bifurcating Time-Periodic Flow of a Navier-Stokes Liquid Past a Cylinder

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Abstract

We provide general sufficient conditions for the existence and uniqueness of branching out of a time-periodic family of solutions from steady-state solutions to the two-dimensional Navier-Stokes equations in the exterior of a cylinder. By separating the time-independent averaged component of the velocity field from its oscillatory one, we show that the problem can be formulated as a coupled elliptic-parabolic nonlinear system in appropriate and distinct function spaces, with the property that the relevant linearized operators become Fredholm of index 0. In this functional setting, the notorious difficulty of 0 being in the essential spectrum entirely disappears and, in fact, it is even meaningless. Our approach is different and, we believe, more natural and simpler than those proposed by previous authors discussing similar questions. Moreover, the latter all fail, when applied to the problem studied here.

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References

  1. Andronov A.A., Witt A.: Sur la theórie mathematiques des autooscillations. C. R. Acad. Sci. Paris, 190, 256–258 (1930)

    MATH  Google Scholar 

  2. Babenko K.I.: On the spectrum of a linearized problem on the flow of a viscous incompressible fluid around a body (Russian). Dokl. Akad. Nauk SSSR 262, 64–68 (1982)

    ADS  MathSciNet  Google Scholar 

  3. Babenko K.I.: Periodic solutions of a problem of the flow of a viscous fluid around a body. Soviet Math. Dokl. 25, 211–216 (1982)

    MathSciNet  MATH  Google Scholar 

  4. Babenko, K.I.: On properties of steady viscous incompressible fluid flows. In: Approximation methods for Navier-Stokes problems (Proc. Sympos., Univ. Paderborn, Paderborn, 1979), Lecture Notes in Math., vol. 771, pp. 12–42. Springer, Berlin, 1980

  5. Chossat, P., Iooss, G.: The Couette-Taylor problem, Applied Mathematical Sciences, vol. 102. Springer-Verlag, New York, 1994

  6. Crandall M.G., Rabinowitz P.H.: The Hopf bifurcation theorem in infinite dimensions. Arch. Rational Mech. Anal. 67, 53–72 (1977)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Farwig R., Neustupa J.: Spectral properties in L q of an Oseen operator modelling fluid flow past a rotating body. Tohoku Math. J. 62, 287–309 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fatone L., Gervasio P., Quarteroni A.: Multimodels for incompressible flows. J. Math. Fluid Mech. 2, 126–150 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Galdi, G.P.: On the Oseen boundary value problem in exterior domains. In: The Navier-Stokes equations II–theory and numerical methods (Oberwolfach, 1991), Lecture Notes in Math., vol. 1530. Springer, Berlin, 111–131, 1992

  10. Galdi, G.P.: Stationary Navier-Stokes problem in a two-dimensional exterior domain. In: Stationary Partial Differential Equations. vol. I, Handb. Differ. Equ. North-Holland, Amsterdam, 71–155, 2004

  11. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-state Problems, 2nd edn. Springer Monographs in Mathematics, Springer, New York, 2011

  12. Galdi G.P.: On time-periodic flow of a viscous liquid past a moving cylinder. Arch. Ration. Mech. Anal. 210, 451–498 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Galdi G.P., Rabier P.J.: Sharp existence results for the stationary Navier-Stokes problem in three-dimensional exterior domains. Arch. Rational Mech. Anal. 154, 343–368 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Gerecht D., Rannacher R., Wollner W.: Computational aspects of pseudospectra in hydrodynamic stability analysis. J. Math. Fluid Mech. 14, 661–692 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Gohberg, I., Goldberg, S. Kaashoek, M.A.: Classes of Linear Operators:I. Operator Theory, Advances and Applications, vol.49. Birkhäuser Verlag, Basel, 1990

  16. Hopf E.: Abzweigung einer periodischen Lösung von einer stationären eines Differentialsystems. Ber. Verh. Sächs. Akad. Wiss. Leipzig. Math.-Nat. Kl. 95, 3–22 (1943)

    MathSciNet  MATH  Google Scholar 

  17. Iooss G.: Existence et stabilité de la solution périodiques secondaire intervenant dans les problémes d’evolution du type Navier-Stokes. Arch. Ration. Mech. Anal. 47, 301–329 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  18. Iooss G., Joseph D.D.: Bifurcation and stability of nT-periodic solutions branching from T-periodic solutions at points of resonance. Arch. Ration. Mech. Anal. 66, 135–172 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  19. Iudovich V.I.: The onset of auto-oscillations in a fluid. J. Appl. Math. Mech. 35, 587–603 (1971)

    Article  MathSciNet  Google Scholar 

  20. Joseph D.D., Sattinger D.H.: Bifurcating time periodic solutions and their stability. Arch. Ration. Mech. Anal. 45, 79–109 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kato, T.: Perturbation Theory for Linear Operators. Springer Classics in Mathematics, 1995

  22. Kielhöfer, H.: Bifurcation Theory. An Introduction with Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 156. Springer, New York, 2012

  23. Kobayashi T., Shibata Y.: On the Oseen equation in the three-dimensional exterior domains. Math. Ann. 310, 1–45 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lindtstedt, A.: Beitrag zur Integration der Differentialgleichungen der Störungstheorie. Abh. K. Akad. Wiss. St. Petersburg 31(4), (1882)

  25. Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1. Springer-Verlag, Berlin Heidelberg New York, 1972

  26. Melcher A., Schneider G., Uecker H.: A Hopf-bifurcation theorem for the vorticity formulation of the Navier-Stokes equations in \({\mathbb{R}^3}\). Commun. Partial Differ. Equ. 33, 772–783 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Poincaré, H.: Les Methodes Nouvelles de la Mecanique Celeste, Vol. I. Paris, 1892

  28. Sazonov L.I.: The onset of auto-oscillations in a flow. Siberian Math. J. 35, 1202–1209 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sazonov L.I.: Justification of the linearization method in the flow problem. Russian Acad. Sci. Izv. Math. 45, 315–337 (1995)

    MathSciNet  Google Scholar 

  30. Tritton, D.J.: Physical Fluid Dynamics, 2nd edn. Clarendon Press, Oxford, 1988

  31. Zeidler, E.: Nonlinear Functional Analysis and Applications, Fixed-Point Theorems, vol.1. Springer-Verlag, New York, 1986

  32. Zeidler, E.: Nonlinear Functional Analysis and Applications, Application to Mathematical Physics, vol.4. Springer-Verlag, New York, 1988

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Correspondence to Giovanni P. Galdi.

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Communicated by P. Rabinowitz

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Galdi, G.P. On Bifurcating Time-Periodic Flow of a Navier-Stokes Liquid Past a Cylinder. Arch Rational Mech Anal 222, 285–315 (2016). https://doi.org/10.1007/s00205-016-1001-3

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