Skip to main content
SpringerLink
Log in

Asymptotic stability of shear-flow solutions to incompressible viscous free boundary problems with and without surface tension

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

This paper concerns the dynamics of a layer of incompressible viscous fluid lying above a rigid plane and with an upper boundary given by a free surface. The fluid is subject to a constant external force with a horizontal component, which arises in modeling the motion of such a fluid down an inclined plane, after a coordinate change. We consider the problem both with and without surface tension for horizontally periodic flows. This problem gives rise to shear-flow equilibrium solutions, and the main thrust of this paper is to study the asymptotic stability of the equilibria in certain parameter regimes. We prove that there exists a parameter regime in which sufficiently small perturbations of the equilibrium at time \(t=0\) give rise to global-in-time solutions that return to equilibrium exponentially in the case with surface tension and almost exponentially in the case without surface tension. We also establish a vanishing surface tension limit, which connects the solutions with and without surface tension.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abergel, F., Bailly, J.-H.: Stationary free surface viscous flows without surface tension in three dimensions. Differential Integral Equations 25(9–10), 801–820 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Abergel, F., Bona, J.: A mathematical theory for viscous, free-surface flows over a perturbed plane. Arch. Rational Mech. Anal. 118(1), 71–93 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beale, J.: The initial value problem for the Navier-Stokes equations with a free surface. Comm. Pure Appl. Math. 34(3), 359–392 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beale, J.: Large-time regularity of viscous surface waves. Arch. Ration. Mech. Anal. 84(4), 307–352 (1983/84)

  5. Beale, J., Nishida, T.: Large-time behavior of viscous surface waves. Recent topics in nonlinear PDE, II (Sendai, 1984), pp. 1–14. North-Holland Math. Stud., 128, North-Holland, Amsterdam (1985)

  6. Danchin, R.: Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients. Rev. Mat. Iberoamericana 21(3), 863–888 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Danchin, R.: Fourier analysis methods for PDEs. Preprint (2005). Lab. d’Analyse et de Math. Appliques, UMR 8050, http://perso-math.univ-mlv.fr/users/danchin.raphael/cours/courschine.pdf

  8. Grenier, E., Guo, Y., Nguyen, T.: Spectral instability of general symmetric shear flows in a two-dimensional channel. Adv. Math. 292, 52–110 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Guo, Y., Tice, I.: Almost exponential decay of periodic viscous surface waves without surface tension. Arch. Ration. Mech. Anal. 207(2), 459–531 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guo, Y., Tice, I.: Decay of viscous surface waves without surface tension in horizontally infinite domains. Anal. PDE 6(6), 1429–1533 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hataya, Y.: Decaying solution of a Navier-Stokes flow without surface tension. J. Math. Kyoto Univ. 49(4), 691–717 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Heisenberg, W.: Über Stabilitt und Turbulenz von Flüssigkeitsströmen. Ann. Phys. 74, 577–627 (1924)

    Article  Google Scholar 

  13. Jang, J., Tice, I., Wang, Y.: The compressible viscous surface-internal wave problem: stability and vanishing surface tension limit. Commun. Math. Phys. 343(3), 1039–1113 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lin, C.: On the stability of two-dimensional parallel flows. Proc. Nat. Acad. Sci. USA 30, 316–323 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  15. Nishida, T., Teramoto, Y., Yoshihara, H.: Global in time behavior of viscous surface waves: horizontally periodic motion. J. Math. Kyoto Univ. 44(2), 271–323 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nishida, T., Teramoto, Y., Yoshihara, H.: Hopf bifurcation in viscous incompressible flow down an inclined plane. J. Math. Fluid Mech. 7(1), 29–71 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Padula, M.: On stability of a capillary liquid down an inclined plane. Discrete Contin. Dyn. Syst. Ser. S 6(5), 1343–1353 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Padula, M.: Nonlinear stability of liquid flow down an inclined plane. Preprint (2012). http://eprints.unife.it/350/

  19. Orr, W.: Stability and instability of steady motions of a perfect liquid and of a viscous fluid, part I. Proc. Ir. Acad. Sect. A 27, 9–68 (1907)

    Google Scholar 

  20. Pileckas, K., Solonnikov, V.: Viscous incompressible free-surface flow down an inclined perturbed plane. Ann. Univ. Ferrara Sez. VII. Sci. Mat. 60(1), 225–244 (2014)

    MathSciNet  MATH  Google Scholar 

  21. Sommerfeld, A.: Ein Beitrag zur hydrodynamischen Erklärung der turbulent Flussigkeitsbewegung. Int. Math. Roma 3, 116–124 (1908)

    Google Scholar 

  22. Sun, S.: On stability of liquid flow down an inclined plane. Quart. Appl. Math. 55(2), 375–397 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tan, Z., Wang, Y.: Zero surface tension limit of viscous surface waves. Commun. Math. Phys. 328(2), 733–807 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Tani, A., Tanaka, N.: Large-time existence of surface waves in incompressible viscous fluids with or without surface tension. Arch. Ration. Mech. Anal. 130(4), 303–314 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Teramoto, Y., Tomoeda, K.: On a linearized system arising in the study of viscous surface flow down an inclined plane. Ann. Univ. Ferrara Sez. VII Sci. Mat. 60(1), 289–306 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tollmien, W.: Asymptotische Integration der Störungsdifferentialgleichung ebener laminarer Strömungen bei hohen Reynoldsschen Zahlen. Z. Angew. Math. Mech. 25/27, 33–50, 70–83 (1947)

  27. Ueno, H., Shiraishi, A., Iguchi, T.: Uniform estimates for the flow of a viscous incompressible fluid down an inclined plane in the thin film regime. J. Math. Anal. Appl. 436(1), 248–287 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wu, L.: Well-posedness and decay of the viscous surface wave. SIAM J. Math. Anal. 46(3), 2084–2135 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Ian Tice

Authors
  1. Ian Tice
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ian Tice.

Additional information

I. Tice was supported by a Simons Foundation Grant (#401468) and an NSF CAREER Grant (DMS #1653161). This work was initiated at the Institute for Computational and Experimental Research in Mathematics (ICERM) during the Spring 2017 semester program “Singularities and Waves In Incompressible Fluids,” which was supported by an NSF Grant (DMS #1439786).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tice, I. Asymptotic stability of shear-flow solutions to incompressible viscous free boundary problems with and without surface tension. Z. Angew. Math. Phys. 69, 28 (2018). https://doi.org/10.1007/s00033-018-0926-9

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-018-0926-9

Keywords

Mathematics Subject Classification

Search

Navigation