Well-posedness of water wave model with viscous effects
Authors:
Rafael Granero-Belinchón and Stefano Scrobogna
Journal:
Proc. Amer. Math. Soc. 148 (2020), 5181-5191
MSC (2010):
Primary 35Q35, 35R35, 35Q31, 35L25
DOI:
https://doi.org/10.1090/proc/15219
Published electronically:
September 18, 2020
MathSciNet review:
4163831
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Abstract | References | Similar Articles | Additional Information
Abstract: Starting from the paper by Dias, Dyachenko, and Zakharov (Physics Letters A, 2008) on viscous water waves, we derive a model that describes water waves with viscosity moving in deep water with or without surface tension effects. This equation takes the form of a nonlocal fourth order wave equation and retains the main contributions to the dynamics of the free surface. Then, we prove the well-posedness in Sobolev spaces of such an equation.
- Benjamin Akers and Paul A. Milewski, Dynamics of three-dimensional gravity-capillary solitary waves in deep water, SIAM J. Appl. Math. 70 (2010), no. 7, 2390–2408. MR 2678044, DOI https://doi.org/10.1137/090758386
- Benjamin Akers and David P. Nicholls, Traveling waves in deep water with gravity and surface tension, SIAM J. Appl. Math. 70 (2010), no. 7, 2373–2389. MR 2678043, DOI https://doi.org/10.1137/090771351
- David M. Ambrose, Jerry L. Bona, and David P. Nicholls, Well-posedness of a model for water waves with viscosity, Discrete Contin. Dyn. Syst. Ser. B 17 (2012), no. 4, 1113–1137. MR 2899939, DOI https://doi.org/10.3934/dcdsb.2012.17.1113
- J. Boussinesq, Lois de l’extinction de la houle en haute mer, C. R. Acad. Sci. Paris 121 (1895), no. 2, 15–20.
- C. H. Arthur Cheng, Rafael Granero-Belinchón, and Steve Shkoller, Well-posedness of the Muskat problem with $H^2$ initial data, Adv. Math. 286 (2016), 32–104. MR 3415681, DOI https://doi.org/10.1016/j.aim.2015.08.026
- C. H. Arthur Cheng, Rafael Granero-Belinchón, Steve Shkoller, and Jon Wilkening, Rigorous asymptotic models of water waves, Water Waves 1 (2019), no. 1, 71–130.
- W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys. 108 (1993), no. 1, 73–83. MR 1239970, DOI https://doi.org/10.1006/jcph.1993.1164
- L. Dawson, H. McGahagan, and G. Ponce, On the decay properties of solutions to a class of Schrödinger equations, Proc. Amer. Math. Soc. 136 (2008), no. 6, 2081–2090. MR 2383514, DOI https://doi.org/10.1090/S0002-9939-08-09355-6
- Frederic Dias, Alexander I. Dyachenko, and Vladimir E. Zakharov, Theory of weakly damped free-surface flows: a new formulation based on potential flow solutions, Phys. Lett. A 372 (2008), no. 8, 1297–1302.
- Denys Dutykh, Visco-potential free-surface flows and long wave modelling, Eur. J. Mech. B Fluids 28 (2009), no. 3, 430–443. MR 2513897, DOI https://doi.org/10.1016/j.euromechflu.2008.11.003
- Denys Dutykh and Frédéric Dias, Dissipative Boussinesq equations, C. R. Mecanique 335 (2007), no. 9–10, 559–583.
- Denys Dutykh and Frédéric Dias, Viscous potential free-surface flows in a fluid layer of finite depth, C. R. Math. Acad. Sci. Paris 345 (2007), no. 2, 113–118 (English, with English and French summaries). MR 2343563, DOI https://doi.org/10.1016/j.crma.2007.06.007
- Denys Dutykh and Olivier Goubet, Derivation of dissipative Boussinesq equations using the Dirichlet-to-Neumann operator approach, Math. Comput. Simulation 127 (2016), 80–93. MR 3501289, DOI https://doi.org/10.1016/j.matcom.2013.12.008
- Loukas Grafakos and Seungly Oh, The Kato-Ponce inequality, Comm. Partial Differential Equations 39 (2014), no. 6, 1128–1157. MR 3200091, DOI https://doi.org/10.1080/03605302.2013.822885
- Rafael Granero-Belinchón and Stefano Scrobogna, On an asymptotic model for free boundary darcy flow in porous media, preprint, arXiv:1810.11798, 2018.
- Rafael Granero-Belinchón and Stefano Scrobogna, Asymptotic models for free boundary flow in porous media, Phys. D 392 (2019), 1–16. MR 3928312, DOI https://doi.org/10.1016/j.physd.2019.02.013
- Rafael Granero-Belinchón and Stefano Scrobogna, Models for damped water waves, SIAM J. Appl. Math. 79 (2019), no. 6, 2530–2550. MR 4041720, DOI https://doi.org/10.1137/19M1262899
- Rafael Granero-Belinchón and Stefano Scrobogna, Well-posedness of the water-wave with viscosity problem, preprint, arXiv:2003.11454, 2020, submitted.
- Rafael Granero-Belinchón and Steve Shkoller, A model for Rayleigh-Taylor mixing and interface turnover, Multiscale Model. Simul. 15 (2017), no. 1, 274–308. MR 3612171, DOI https://doi.org/10.1137/16M1083463
- Lei Jiang, Chao-Lung Ting, Marc Perlin, and William W. Schultz, Moderate and steep Faraday waves: instabilities, modulation and temporal asymmetries, J. Fluid Mech. 329 (1996), 275–307.
- D. D. Joseph and J. Wang, The dissipation approximation and viscous potential flow, J. Fluid Mech. 505 (2004), 365–377. MR 2259003, DOI https://doi.org/10.1017/S0022112004008602
- Maria Kakleas and David P. Nicholls, Numerical simulation of a weakly nonlinear model for water waves with viscosity, J. Sci. Comput. 42 (2010), no. 2, 274–290. MR 2578037, DOI https://doi.org/10.1007/s10915-009-9324-y
- Tosio Kato and Gustavo Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907. MR 951744, DOI https://doi.org/10.1002/cpa.3160410704
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), no. 4, 527–620. MR 1211741, DOI https://doi.org/10.1002/cpa.3160460405
- C. Skandrani, C. Kharif, and J. Poitevin, Nonlinear evolution of water surface waves: the frequency down-shift phenomenon, Mathematical problems in the theory of water waves (Luminy, 1995) Contemp. Math., vol. 200, Amer. Math. Soc., Providence, RI, 1996, pp. 157–171. MR 1410506, DOI https://doi.org/10.1090/conm/200/02514
- Horace Lamb, Hydrodynamics, 6th ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993. With a foreword by R. A. Caflisch [Russel E. Caflisch]. MR 1317348
- Michael S. Longuet-Higgins, Theory of weakly damped Stokes waves: a new formulation and its physical interpretation, J. Fluid Mech. 235 (1992), 319–324. MR 1150193, DOI https://doi.org/10.1017/S0022112092001125
- Y Matsuno Nonlinear evolutions of surface gravity waves on fluid of finite depth. Physical review letters, 69(4):609, 1992.
- Y. Matsuno, Nonlinear evolution of surface gravity waves over an uneven bottom, J. Fluid Mech. 249 (1993), 121–133. MR 1214472, DOI https://doi.org/10.1017/S0022112093001107
- Yoshimasa Matsuno, Two-dimensional evolution of surface gravity waves on a fluid of arbitrary depth, Physical Review E, 47(6):4593, 1993.
- Marième Ngom and David P. Nicholls, Well-posedness and analyticity of solutions to a water wave problem with viscosity, J. Differential Equations 265 (2018), no. 10, 5031–5065. MR 3848244, DOI https://doi.org/10.1016/j.jde.2018.06.030
- K. D. Ruvinsky, F. I. Feldstein, and G. I. Freidman, Numerical simulations of the quasi-stationary stage of ripple excitation by steep gravity–capillary waves, J. Fluid Mech. 230 (1991), 339–353.
- K. D. Ruvinsky and G. I. Freidman, Improvement of the first Stokes method for the investigation of finite-amplitude potential gravity-capillary waves, IX All-Union Symp. on Diffraction and Propagation Waves, Tbilisi: Theses of Reports, volume 2, 1985, pp. 22–25.
- K. D. Ruvinsky and G. I. Freidman. The fine structure of strong gravity-capillary waves, Nonlinear waves: Structures and Bifurcations, AV Gaponov-Grekhov and MI Rabinovich, eds. Moscow: Nauka, pages 304–326, 1987.
- George Gabriel Stokes, On the Theory of Oscillatory Waves, volume 1 of Cambridge Library Collection - Mathematics, Cambridge University Press, 1847.
- J. Wang and D. D. Joseph, Purely irrotational theories of the effect of the viscosity on the decay of free gravity waves, J. Fluid Mech. 559 (2006), 461–472. MR 2266175, DOI https://doi.org/10.1017/S0022112006000401
- Guangyu Wu, Yuming Liu, and Dick K. P. Yue, A note on stabilizing the Benjamin-Feir instability, J. Fluid Mech. 556 (2006), 45–54. MR 2263443, DOI https://doi.org/10.1017/S0022112005008293
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Additional Information
Rafael Granero-Belinchón
Affiliation:
Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avenida. Los Castros s/n, Santander, Spain
ORCID:
0000-0003-2752-8086
Email:
rafael.granero@unican.es
Stefano Scrobogna
Affiliation:
IMUS, University of Seville, 41012 Sevilla, Spain; and Basque Center of Applied Mathematics, Mazarredo 14, 48009, Bilbao, Spain
MR Author ID:
1229361
Email:
sscrobogna@bcamath.org
Received by editor(s):
November 5, 2019
Received by editor(s) in revised form:
March 31, 2020
Published electronically:
September 18, 2020
Additional Notes:
The first author was funded by the grant MTM2017-89976-P from the Spanish government.
The research of the second author was supported by the Basque Government through the BERC 2018-2021 program and by the Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and through project MTM2017-82184-R funded by (AEI/FEDER, UE) and “DESFLU” and by the European Research Council through the Starting Grant project H2020-EU.1.1.-639227 FLUID-INTERFACE
Communicated by:
Catherine Sulem
Article copyright:
© Copyright 2020
American Mathematical Society