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.
Similar content being viewed by others
References
Andronov A.A., Witt A.: Sur la theórie mathematiques des autooscillations. C. R. Acad. Sci. Paris, 190, 256–258 (1930)
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)
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)
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
Chossat, P., Iooss, G.: The Couette-Taylor problem, Applied Mathematical Sciences, vol. 102. Springer-Verlag, New York, 1994
Crandall M.G., Rabinowitz P.H.: The Hopf bifurcation theorem in infinite dimensions. Arch. Rational Mech. Anal. 67, 53–72 (1977)
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)
Fatone L., Gervasio P., Quarteroni A.: Multimodels for incompressible flows. J. Math. Fluid Mech. 2, 126–150 (2000)
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
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
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
Galdi G.P.: On time-periodic flow of a viscous liquid past a moving cylinder. Arch. Ration. Mech. Anal. 210, 451–498 (2013)
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)
Gerecht D., Rannacher R., Wollner W.: Computational aspects of pseudospectra in hydrodynamic stability analysis. J. Math. Fluid Mech. 14, 661–692 (2012)
Gohberg, I., Goldberg, S. Kaashoek, M.A.: Classes of Linear Operators:I. Operator Theory, Advances and Applications, vol.49. Birkhäuser Verlag, Basel, 1990
Hopf E.: Abzweigung einer periodischen Lösung von einer stationären eines Differentialsystems. Ber. Verh. Sächs. Akad. Wiss. Leipzig. Math.-Nat. Kl. 95, 3–22 (1943)
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)
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)
Iudovich V.I.: The onset of auto-oscillations in a fluid. J. Appl. Math. Mech. 35, 587–603 (1971)
Joseph D.D., Sattinger D.H.: Bifurcating time periodic solutions and their stability. Arch. Ration. Mech. Anal. 45, 79–109 (1972)
Kato, T.: Perturbation Theory for Linear Operators. Springer Classics in Mathematics, 1995
Kielhöfer, H.: Bifurcation Theory. An Introduction with Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 156. Springer, New York, 2012
Kobayashi T., Shibata Y.: On the Oseen equation in the three-dimensional exterior domains. Math. Ann. 310, 1–45 (1998)
Lindtstedt, A.: Beitrag zur Integration der Differentialgleichungen der Störungstheorie. Abh. K. Akad. Wiss. St. Petersburg 31(4), (1882)
Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1. Springer-Verlag, Berlin Heidelberg New York, 1972
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)
Poincaré, H.: Les Methodes Nouvelles de la Mecanique Celeste, Vol. I. Paris, 1892
Sazonov L.I.: The onset of auto-oscillations in a flow. Siberian Math. J. 35, 1202–1209 (1994)
Sazonov L.I.: Justification of the linearization method in the flow problem. Russian Acad. Sci. Izv. Math. 45, 315–337 (1995)
Tritton, D.J.: Physical Fluid Dynamics, 2nd edn. Clarendon Press, Oxford, 1988
Zeidler, E.: Nonlinear Functional Analysis and Applications, Fixed-Point Theorems, vol.1. Springer-Verlag, New York, 1986
Zeidler, E.: Nonlinear Functional Analysis and Applications, Application to Mathematical Physics, vol.4. Springer-Verlag, New York, 1988
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Rabinowitz
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-016-1001-3