The nonlinear stability regime of the viscous Faraday wave problem
Authors:
David Altizio, Ian Tice, Xinyu Wu and Taisuke Yasuda
Journal:
Quart. Appl. Math. 78 (2020), 545-587
MSC (2010):
Primary 35Q30, 35R35, 76E17; Secondary 35B40, 76D45
DOI:
https://doi.org/10.1090/qam/1562
Published electronically:
December 18, 2019
MathSciNet review:
4148819
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Additional Information
Abstract: This paper concerns the dynamics of a layer of incompressible viscous fluid lying above a vertically oscillating rigid plane and with an upper boundary given by a free surface. We consider the problem with gravity and surface tension for horizontally periodic flows. This problem gives rise to flat but vertically oscillating equilibrium solutions, and the main thrust of this paper is to study the asymptotic stability of these equilibria in certain parameter regimes. We prove that both with and without surface tension 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 decay to equilibrium at an identified quantitative rate.
References
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- Zhong Tan and Yanjin Wang, Zero surface tension limit of viscous surface waves, Comm. Math. Phys. 328 (2014), no. 2, 733–807. MR 3199998, DOI https://doi.org/10.1007/s00220-014-1986-0
- Zhong Tan and Yanjin Wang, Global well-posedness of an initial-boundary value problem for viscous non-resistive MHD systems, SIAM J. Math. Anal. 50 (2018), no. 1, 1432–1470. MR 3766969, DOI https://doi.org/10.1137/16M1088156
- Atusi Tani and Naoto Tanaka, Large-time existence of surface waves in incompressible viscous fluids with or without surface tension, Arch. Rational Mech. Anal. 130 (1995), no. 4, 303–314. MR 1346360, DOI https://doi.org/10.1007/BF00375142
- Ian Tice, Asymptotic stability of shear-flow solutions to incompressible viscous free boundary problems with and without surface tension, Z. Angew. Math. Phys. 69 (2018), no. 2, Paper No. 28, 39. MR 3764545, DOI https://doi.org/10.1007/s00033-018-0926-9
- Yanjin Wang, Ian Tice, and Chanwoo Kim, The viscous surface-internal wave problem: global well-posedness and decay, Arch. Ration. Mech. Anal. 212 (2014), no. 1, 1–92. MR 3162473, DOI https://doi.org/10.1007/s00205-013-0700-2
- Mark-Tiele Westra, Doug J. Binks, and Willem van de Water, Patterns of Faraday waves, J. Fluid Mech. 496 (2003), 1–32. MR 2029259, DOI https://doi.org/10.1017/S0022112003005895
- Lei Wu, Well-posedness and decay of the viscous surface wave, SIAM J. Math. Anal. 46 (2014), no. 3, 2084–2135. MR 3223930, DOI https://doi.org/10.1137/120897018
References
- J. Thomas Beale, The initial value problem for the Navier-Stokes equations with a free surface, Comm. Pure Appl. Math. 34 (1981), no. 3, 359–392. MR 611750, DOI https://doi.org/10.1002/cpa.3160340305
- J. T. Beale, Large-time regularity of viscous surface waves, Arch. Rational Mech. Anal. 84 (1983/84), no. 4, 307–352. MR 721189, DOI https://doi.org/10.1007/BF00250586
- J. Thomas Beale and Takaaki Nishida, Large-time behavior of viscous surface waves, Recent topics in nonlinear PDE, II (Sendai, 1984) North-Holland Math. Stud., vol. 128, North-Holland, Amsterdam, 1985, pp. 1–14. MR 882925, DOI https://doi.org/10.1016/S0304-0208%2808%2972355-7
- T. B. Benjamin and F. Ursell, The stability of the plane free surface of a liquid in vertical periodic motion, Proc. Roy. Soc. London Ser. A 225 (1954), 505–515. MR 65315, DOI https://doi.org/10.1098/rspa.1954.0218
- John W. M. Bush, Pilot-wave hydrodynamics, Annual review of fluid mechanics. Vol. 47, Annu. Rev. Fluid Mech., vol. 47, Annual Reviews, Palo Alto, CA, 2015, pp. 269–292. MR 3727170
- Y. Couder, S. Protiere, E. Fort, and A. Boudaoud, Dynamical phenomena: Walking and orbiting droplets, Nature 437(7056) (2005), 208.
- Raphaël Danchin, Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients, Rev. Mat. Iberoamericana 21 (2005), no. 3, 863–888. MR 2231013, DOI https://doi.org/10.4171/RMI/438
- Kausik S. Das, Stephen W. Morris, and A. Bhattacharyay, Parametric internal waves in a compressible fluid, Nonlinearity 22 (2009), no. 12, 2981–2990. MR 2565359, DOI https://doi.org/10.1088/0951-7715/22/12/010
- M. Faraday, On the forms and states assumed by fluids in contact with vibrating elastic surfaces, Philos. Trans. R. Soc. London 121 (1831), 39–346.
- Yan Guo and Ian Tice, Almost exponential decay of periodic viscous surface waves without surface tension, Arch. Ration. Mech. Anal. 207 (2013), no. 2, 459–531. MR 3005322, DOI https://doi.org/10.1007/s00205-012-0570-z
- Yan Guo and Ian Tice, Decay of viscous surface waves without surface tension in horizontally infinite domains, Anal. PDE 6 (2013), no. 6, 1429–1533. MR 3148059, DOI https://doi.org/10.2140/apde.2013.6.1429
- Yasushi Hataya, Decaying solution of a Navier-Stokes flow without surface tension, J. Math. Kyoto Univ. 49 (2009), no. 4, 691–717. MR 2591112, DOI https://doi.org/10.1215/kjm/1265899478
- Juhi Jang, Ian Tice, and Yanjin Wang, The compressible viscous surface-internal wave problem: Stability and vanishing surface tension limit, Comm. Math. Phys. 343 (2016), no. 3, 1039–1113. MR 3488552, DOI https://doi.org/10.1007/s00220-016-2603-1
- Krishna Kumar, Linear theory of Faraday instability in viscous liquids, Proc. Roy. Soc. London Ser. A 452 (1996), no. 1948, 1113–1126. MR 1396518, DOI https://doi.org/10.1098/rspa.1996.0056
- Krishna Kumar and Laurette S. Tuckerman, Parametric instability of the interface between two fluids, J. Fluid Mech. 279 (1994), 49–68. MR 1306814, DOI https://doi.org/10.1017/S0022112094003812
- N. W. McLachlan, Theory and application of Mathieu functions, Dover Publications, Inc., New York, 1964. MR 0174808
- John Miles and Diane Henderson, Parametrically forced surface waves, Annual review of fluid mechanics, Vol. 22, Annual Reviews, Palo Alto, CA, 1990, pp. 143–165. MR 1043919
- Takaaki Nishida, Yoshiaki Teramoto, and Hideaki Yoshihara, Global in time behavior of viscous surface waves: Horizontally periodic motion, J. Math. Kyoto Univ. 44 (2004), no. 2, 271–323. MR 2081074, DOI https://doi.org/10.1215/kjm/1250283555
- Nicolas Périnet, Damir Juric, and Laurette S. Tuckerman, Numerical simulation of Faraday waves, J. Fluid Mech. 635 (2009), 1–26. MR 2540462, DOI https://doi.org/10.1017/S0022112009007551
- Saad Qadeer, Simulating Nonlinear Faraday Waves on a Cylinder, ProQuest LLC, Ann Arbor, MI, 2018. Thesis (Ph.D.)–University of California, Berkeley. MR 3864883
- A. Remond-Tiedrez and I. Tice, The viscous surface wave problem with generalized surface energies, to appear in SIAM J. Math. Anal., arXiv:1806.07660 (2019).
- A. Skeldon and A. Rucklidge, Can weakly nonlinear theory explain Faraday wave patterns near onset?, Journal of Fluid Mechanics 777 (2015), 604–632.
- Zhong Tan and Yanjin Wang, Zero surface tension limit of viscous surface waves, Comm. Math. Phys. 328 (2014), no. 2, 733–807. MR 3199998, DOI https://doi.org/10.1007/s00220-014-1986-0
- Zhong Tan and Yanjin Wang, Global well-posedness of an initial-boundary value problem for viscous non-resistive MHD systems, SIAM J. Math. Anal. 50 (2018), no. 1, 1432–1470. MR 3766969, DOI https://doi.org/10.1137/16M1088156
- Atusi Tani and Naoto Tanaka, Large-time existence of surface waves in incompressible viscous fluids with or without surface tension, Arch. Rational Mech. Anal. 130 (1995), no. 4, 303–314. MR 1346360, DOI https://doi.org/10.1007/BF00375142
- Ian Tice, Asymptotic stability of shear-flow solutions to incompressible viscous free boundary problems with and without surface tension, Z. Angew. Math. Phys. 69 (2018), no. 2, Art. 28, 39. MR 3764545, DOI https://doi.org/10.1007/s00033-018-0926-9
- Yanjin Wang, Ian Tice, and Chanwoo Kim, The viscous surface-internal wave problem: Global well-posedness and decay, Arch. Ration. Mech. Anal. 212 (2014), no. 1, 1–92. MR 3162473, DOI https://doi.org/10.1007/s00205-013-0700-2
- Mark-Tiele Westra, Doug J. Binks, and Willem van de Water, Patterns of Faraday waves, J. Fluid Mech. 496 (2003), 1–32. MR 2029259, DOI https://doi.org/10.1017/S0022112003005895
- Lei Wu, Well-posedness and decay of the viscous surface wave, SIAM J. Math. Anal. 46 (2014), no. 3, 2084–2135. MR 3223930, DOI https://doi.org/10.1137/120897018
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Additional Information
David Altizio
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email:
altizio2@illinois.edu
Ian Tice
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
MR Author ID:
830666
Email:
iantice@andrew.cmu.edu
Xinyu Wu
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
MR Author ID:
1309671
Email:
xinyuw1@andrew.cmu.edu
Taisuke Yasuda
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email:
taisukey@andrew.cmu.edu
Keywords:
Faraday waves,
free boundary problems,
asymptotic stability
Received by editor(s):
May 16, 2019
Received by editor(s) in revised form:
September 30, 2019
Published electronically:
December 18, 2019
Additional Notes:
The second author was supported by an NSF CAREER Grant (DMS #1653161). The first, third, and fourth authors were supported by the summer research support provided by this grant.
Article copyright:
© Copyright 2019
Brown University
