Attainability of time-periodic flow of a viscous liquid past an oscillating body


A body \(\mathscr {B}\) is started from rest by a translational motion in an otherwise quiescent Navier–Stokes liquid filling the whole space. We show, for small data, that if after some time \(\mathscr {B}\) reaches a spinless oscillatory motion of period \(\mathcal T\), the liquid will eventually execute also a time periodic motion with the same period \(\mathcal T\). This result is a suitable generalization of the famous Finn’s starting problem for steady states, to the case of time-periodic motions.

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  1. 1.

    J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, 1976.

    Google Scholar 

  2. 2.

    R. Finn, Stationary solutions of the Navier-Stokes equations, Proc. Symp. Appl. Math. 17 (1965), 121–153.

    Article  Google Scholar 

  3. 3.

    G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady-State Problems, Second Edition, Springer, 2011.

    Google Scholar 

  4. 4.

    G.P. Galdi, Viscous flow past a body translating by time-periodic motion with zero average, Arch. Rational Mech. Anal. 237 (2020), 1237–1269.

    MathSciNet  Article  Google Scholar 

  5. 5.

    G.P. Galdi, J.G. Heywood and Y. Shibata, On the global existence and convergence to steady state of Navier-Stokes flow past an obstacle that is started from rest, Arch. Rational Mech. Anal. 138 (1997), 307–318.

    MathSciNet  Article  Google Scholar 

  6. 6.

    T. Hansel and A. Rhandi, The Oseen-Navier-Stokes flow in the exterior of a rotating obstacle: the non-autonomous case, J. Reine Angew. Math. 694 (2014), 1–26.

    MathSciNet  Article  Google Scholar 

  7. 7.

    T. Hishida, Large time behavior of a generalized Oseen evolution operator, with applications to the Navier-Stokes flow past a rotating obstacle, Math. Ann. 372 (2018), 915–949.

    MathSciNet  Article  Google Scholar 

  8. 8.

    T. Hishida, Decay estimates of gradient of a generalized Oseen evolution operator arising from time-dependent rigid motions in exterior domains, Arch. Rational Mech. Anal. 238 (2020), 215–254.

    MathSciNet  Article  Google Scholar 

  9. 9.

    T. Hishida and Y. Shibata, \(L_p\)-\(L_q\) estimate of the Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle, Arch. Rational Mech. Anal. 193 (2009), 339–421.

    MathSciNet  Article  Google Scholar 

  10. 10.

    T. Hishida, and P. Maremonti, Navier–Stokes flow past a rigid body: attainability of steady solutions as limits of unsteady weak solutions, starting and landing cases. J. Math. Fluid Mech. 20 (2018) 771–800

    MathSciNet  Article  Google Scholar 

  11. 11.

    H. Iwashita, \(L_q\)-\(L_r\) estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in \(L_q\) spaces, Math. Ann. 285 (1989), 265–288.

    MathSciNet  Article  Google Scholar 

  12. 12.

    H. Koba, On \(L^{3,\infty }\)-stability of the Navier-Stokes system in exterior domains, J. Differ. Equ. 262 (2017), 2618–2683.

    Article  Google Scholar 

  13. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 14.

    P. Maremonti and V.A. Solonnikov, On nonstationary Stokes problems in exterior domains, Ann. Sc. Norm. Sup. Pisa 24 (1997), 395–449.

    MathSciNet  MATH  Google Scholar 

  15. 15.

    T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J. 12 (1982), 115–140.

    MathSciNet  Article  Google Scholar 

  16. 16.

    C.G. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in \(L^q\)-spaces for bounded and exterior domains, Mathematical Problems Relating to the Navier-Stokes Equations (eds. G.P. Galdi), 1–35, Ser. Adv. Math. Appl. Sci. 11, World Sci. Publ., River Edge, NJ, 1992.

  17. 17.

    H. Tanabe, Equations of Evolution, Pitman, London, 1979.

    Google Scholar 

  18. 18.

    M. Yamazaki, The Navier-Stokes equations in the weak-\(L^n\) space with time-dependent external force, Math. Ann. 317 (2000), 635–675.

    MathSciNet  Article  Google Scholar 

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Correspondence to Giovanni P. Galdi.

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G. P. Galdi: Partially supported by NSF Grant DMS-1614011.

T. Hishida: Partially supported by the Grant-in-Aid for Scientific Research 18K03363 from JSPS.

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Galdi, G.P., Hishida, T. Attainability of time-periodic flow of a viscous liquid past an oscillating body. J. Evol. Equ. (2021).

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