American Institute of Mathematical Sciences

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June  2019, 39(6): 3123-3147. doi: 10.3934/dcds.2019129

Sufficiently strong dispersion removes ill-posedness in truncated series models of water waves

Shunlian Liu 1, and  David M. Ambrose 2,,

1. 

School of Science, Hunan University of Technology, Zhuzhou, Hunan 412007, China
2. 

Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA

* Corresponding author: David M. Ambrose

Received  April 2018 Revised  August 2018 Published  February 2019

Fund Project: The second author is grateful for support from the National Science Foundation through grant DMS-1515849.

Truncated series models of gravity water waves are popular for use in simulation. Recent work has shown that these models need not inherit the well-posedness properties of the full equations of motion (the irrotational, incompressible Euler equations). We show that if one adds a sufficiently strong dispersive term to a quadratic truncated series model, the system then has a well-posed initial value problem. Such dispersion can be relevant in certain physical contexts, such as in the case of a bending force present at the free surface, as in a hydroelastic sheet.

Keywords: Water waves, hydroelastic, operator expansions, paradifferential calculus, well-posedness.
Mathematics Subject Classification: Primary: 76B15; Secondary: 34A12, 76B07, 35S50.
Citation: Shunlian Liu, David M. Ambrose. Sufficiently strong dispersion removes ill-posedness in truncated series models of water waves. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3123-3147. doi: 10.3934/dcds.2019129
References:
[1]

T. Alazard, N. Burq and C. Zuily, On the water-wave equations with surface tension, Duke Math. J., 158 (2011), 413-499.  doi: 10.1215/00127094-1345653.  Google Scholar

[2]

D. Ambrose, J. Bona and D. Nicholls, Well-posedness of a model for water waves with viscosity, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1113-1137.  doi: 10.3934/dcdsb.2012.17.1113.  Google Scholar

[3]

D. Ambrose, J. Bona and D. Nicholls, On ill-posedness of truncated series models for water waves, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20130849, 16pp. doi: 10.1098/rspa.2013.0849.  Google Scholar

[4]

D. Ambrose and N. Masmoudi, The zero surface tension limit of two-dimensional water waves, Comm. Pure Appl. Math., 58 (2005), 1287-1315.  doi: 10.1002/cpa.20085.  Google Scholar

[5]

D. Ambrose and N. Masmoudi, The zero surface tension limit of three-dimensional water waves, Indiana Univ. Math. J., 58 (2009), 479-521.  doi: 10.1512/iumj.2009.58.3450.  Google Scholar

[6]

D. Ambrose and M. Siegel, Well-posedness of two-dimensional hydroelastic waves, Proc. Roy. Soc. Edinburgh Sect. A, 147 (2017), 529-570.  doi: 10.1017/S0308210516000238.  Google Scholar

[7]

D. Ambrose and M. Siegel, Ill-posedness of quadratic truncated series models of gravity water waves, Preprint. Google Scholar

[8]

D. Ambrose and G. Simpson, Local existence theory for derivative nonlinear Schrödinger equations with noninteger power nonlinearities, SIAM J. Math. Anal., 47 (2015), 2241-2264.  doi: 10.1137/140955227.  Google Scholar

[9]

P. Baldi and J. Toland, Bifurcation and secondary bifurcation of heavy periodic hydroelastic travelling waves, Interfaces Free Bound., 12 (2010), 1-22.  doi: 10.4171/IFB/224.  Google Scholar

[10]

T. Benjamin and T. Bridges, Reappraisal of the Kelvin-Helmholtz problem. Ⅱ. Interaction of the Kelvin-Helmholtz, superharmonic and Benjamin-Feir instabilities, J. Fluid Mech., 333 (1997), 327-373.  doi: 10.1017/S0022112096004284.  Google Scholar

[11]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4), 14 (1981), 209-246, URL http://www.numdam.org/item?id=ASENS_1981_4_14_2_209_0.  Google Scholar

[12]

R. Caflisch and O. Orellana, Singular solutions and ill-posedness for the evolution of vortex sheets, SIAM J. Math. Anal., 20 (1989), 293-307.  doi: 10.1137/0520020.  Google Scholar

[13]

H. Christianson, V. Hur and G. Staffilani, Strichartz estimates for the water-wave problem with surface tension, Comm. Partial Differential Equations, 35 (2010), 2195-2252.  doi: 10.1080/03605301003758351.  Google Scholar

[14]

P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439.  doi: 10.2307/1990923.  Google Scholar

[15]

W. Craig, P. Guyenne and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces, Comm. Pure Appl. Math., 58 (2005), 1587-1641.  doi: 10.1002/cpa.20098.  Google Scholar

[16]

W. Craig, P. Guyenne and C. Sulem, Water waves over a random bottom, J. Fluid Mech., 640 (2009), 79-107.  doi: 10.1017/S0022112009991248.  Google Scholar

[17]

W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83.  doi: 10.1006/jcph.1993.1164.  Google Scholar

[18]

A. de Bouard, W. Craig, O. Díaz-Espinosa, P. Guyenne and C. Sulem, Long wave expansions for water waves over random topography, Nonlinearity, 21 (2008), 2143-2178.  doi: 10.1088/0951-7715/21/9/014.  Google Scholar

[19]

F. Dias, A. Dyachenko and V. Zakharov, Theory of weakly damped free-surface flows: A new formulation based on potential flow solutions, Phys. Lett. A, 372 (2008), 1297-1302.   Google Scholar

[20]

M. Groves, B. Hewer and E. Wahlén, Variational existence theory for hydroelastic solitary waves, C. R. Math. Acad. Sci. Paris, 354 (2016), 1078-1086.  doi: 10.1016/j.crma.2016.10.004.  Google Scholar

[21]

P. Guyenne and E. Părău, Computations of fully nonlinear hydroelastic solitary waves on deep water, J. Fluid Mech., 713 (2012), 307-329.  doi: 10.1017/jfm.2012.458.  Google Scholar

[22]

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, vol. 26 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 1997.  Google Scholar

[23]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, in Studies in Applied Mathematics, vol. 8 of Adv. Math. Suppl. Stud., Academic Press, New York, 1983, 93-128.  Google Scholar

[24]

D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654(electronic).  doi: 10.1090/S0894-0347-05-00484-4.  Google Scholar

[25]

D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013, Mathematical analysis and asymptotics. doi: 10.1090/surv/188.  Google Scholar

[26]

S. Liu, Well-posedness of Hydroelastic Waves and Their Truncated Series Models, PhD thesis, Drexel University, 2016.  Google Scholar

[27]

S. Liu and D. Ambrose, Well-posedness of two-dimensional hydroelastic waves with mass, J. Differential Equations, 262 (2017), 4656-4699, URL http://www.sciencedirect.com/science/article/pii/S0022039616304879, In press. doi: 10.1016/j.jde.2016.12.016.  Google Scholar

[28]

G. Métivier, Para-differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems, vol. 5 of Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series, Edizioni della Normale, Pisa, 2008.,  Google Scholar

[29]

D. Milder, The effects of truncation on surface-wave Hamiltonians, J. Fluid Mech., 217 (1990), 249-262.  doi: 10.1017/S0022112090000714.  Google Scholar

[30]

D. Nicholls, Spectral stability of traveling water waves: Eigenvalue collision, singularities, and direct numerical simulation, Phys. D, 240 (2011), 376-381, URL http://www.sciencedirect.com/science/article/pii/S0167278910002630. Google Scholar

[31]

P. Plotnikov and J. Toland, Modelling nonlinear hydroelastic waves, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 369 (2011), 2942-2956.  doi: 10.1098/rsta.2011.0104.  Google Scholar

[32]

M. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000.  doi: 10.1080/03605308208820242.  Google Scholar

[33]

M. Taylor, Pseudodifferential Operators and Nonlinear PDE, vol. 100 of Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1991. doi: 10.1007/978-1-4612-0431-2.  Google Scholar

[34]

J. Toland, Heavy hydroelastic travelling waves, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2371-2397.  doi: 10.1098/rspa.2007.1883.  Google Scholar

[35]

J. Wilkening and V. Vasan, Comparison of five methods of computing the Dirichlet-Neumann operator for the water wave problem, in Nonlinear wave equations: analytic and computational techniques, vol. 635 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2015, 175–210. doi: 10.1090/conm/635/12713.  Google Scholar

[36]

S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D, Invent. Math., 130 (1997), 39-72.  doi: 10.1007/s002220050177.  Google Scholar

[37]

S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445-495.  doi: 10.1090/S0894-0347-99-00290-8.  Google Scholar

[38]

V. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Journal of Applied Mechanics and Technical Physics, 9 (1968), 190-194.  doi: 10.1007/BF00913182.  Google Scholar

show all references

References:
[1]

T. Alazard, N. Burq and C. Zuily, On the water-wave equations with surface tension, Duke Math. J., 158 (2011), 413-499.  doi: 10.1215/00127094-1345653.  Google Scholar

[2]

D. Ambrose, J. Bona and D. Nicholls, Well-posedness of a model for water waves with viscosity, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1113-1137.  doi: 10.3934/dcdsb.2012.17.1113.  Google Scholar

[3]

D. Ambrose, J. Bona and D. Nicholls, On ill-posedness of truncated series models for water waves, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20130849, 16pp. doi: 10.1098/rspa.2013.0849.  Google Scholar

[4]

D. Ambrose and N. Masmoudi, The zero surface tension limit of two-dimensional water waves, Comm. Pure Appl. Math., 58 (2005), 1287-1315.  doi: 10.1002/cpa.20085.  Google Scholar

[5]

D. Ambrose and N. Masmoudi, The zero surface tension limit of three-dimensional water waves, Indiana Univ. Math. J., 58 (2009), 479-521.  doi: 10.1512/iumj.2009.58.3450.  Google Scholar

[6]

D. Ambrose and M. Siegel, Well-posedness of two-dimensional hydroelastic waves, Proc. Roy. Soc. Edinburgh Sect. A, 147 (2017), 529-570.  doi: 10.1017/S0308210516000238.  Google Scholar

[7]

D. Ambrose and M. Siegel, Ill-posedness of quadratic truncated series models of gravity water waves, Preprint. Google Scholar

[8]

D. Ambrose and G. Simpson, Local existence theory for derivative nonlinear Schrödinger equations with noninteger power nonlinearities, SIAM J. Math. Anal., 47 (2015), 2241-2264.  doi: 10.1137/140955227.  Google Scholar

[9]

P. Baldi and J. Toland, Bifurcation and secondary bifurcation of heavy periodic hydroelastic travelling waves, Interfaces Free Bound., 12 (2010), 1-22.  doi: 10.4171/IFB/224.  Google Scholar

[10]

T. Benjamin and T. Bridges, Reappraisal of the Kelvin-Helmholtz problem. Ⅱ. Interaction of the Kelvin-Helmholtz, superharmonic and Benjamin-Feir instabilities, J. Fluid Mech., 333 (1997), 327-373.  doi: 10.1017/S0022112096004284.  Google Scholar

[11]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4), 14 (1981), 209-246, URL http://www.numdam.org/item?id=ASENS_1981_4_14_2_209_0.  Google Scholar

[12]

R. Caflisch and O. Orellana, Singular solutions and ill-posedness for the evolution of vortex sheets, SIAM J. Math. Anal., 20 (1989), 293-307.  doi: 10.1137/0520020.  Google Scholar

[13]

H. Christianson, V. Hur and G. Staffilani, Strichartz estimates for the water-wave problem with surface tension, Comm. Partial Differential Equations, 35 (2010), 2195-2252.  doi: 10.1080/03605301003758351.  Google Scholar

[14]

P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439.  doi: 10.2307/1990923.  Google Scholar

[15]

W. Craig, P. Guyenne and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces, Comm. Pure Appl. Math., 58 (2005), 1587-1641.  doi: 10.1002/cpa.20098.  Google Scholar

[16]

W. Craig, P. Guyenne and C. Sulem, Water waves over a random bottom, J. Fluid Mech., 640 (2009), 79-107.  doi: 10.1017/S0022112009991248.  Google Scholar

[17]

W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83.  doi: 10.1006/jcph.1993.1164.  Google Scholar

[18]

A. de Bouard, W. Craig, O. Díaz-Espinosa, P. Guyenne and C. Sulem, Long wave expansions for water waves over random topography, Nonlinearity, 21 (2008), 2143-2178.  doi: 10.1088/0951-7715/21/9/014.  Google Scholar

[19]

F. Dias, A. Dyachenko and V. Zakharov, Theory of weakly damped free-surface flows: A new formulation based on potential flow solutions, Phys. Lett. A, 372 (2008), 1297-1302.   Google Scholar

[20]

M. Groves, B. Hewer and E. Wahlén, Variational existence theory for hydroelastic solitary waves, C. R. Math. Acad. Sci. Paris, 354 (2016), 1078-1086.  doi: 10.1016/j.crma.2016.10.004.  Google Scholar

[21]

P. Guyenne and E. Părău, Computations of fully nonlinear hydroelastic solitary waves on deep water, J. Fluid Mech., 713 (2012), 307-329.  doi: 10.1017/jfm.2012.458.  Google Scholar

[22]

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, vol. 26 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 1997.  Google Scholar

[23]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, in Studies in Applied Mathematics, vol. 8 of Adv. Math. Suppl. Stud., Academic Press, New York, 1983, 93-128.  Google Scholar

[24]

D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654(electronic).  doi: 10.1090/S0894-0347-05-00484-4.  Google Scholar

[25]

D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013, Mathematical analysis and asymptotics. doi: 10.1090/surv/188.  Google Scholar

[26]

S. Liu, Well-posedness of Hydroelastic Waves and Their Truncated Series Models, PhD thesis, Drexel University, 2016.  Google Scholar

[27]

S. Liu and D. Ambrose, Well-posedness of two-dimensional hydroelastic waves with mass, J. Differential Equations, 262 (2017), 4656-4699, URL http://www.sciencedirect.com/science/article/pii/S0022039616304879, In press. doi: 10.1016/j.jde.2016.12.016.  Google Scholar

[28]

G. Métivier, Para-differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems, vol. 5 of Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series, Edizioni della Normale, Pisa, 2008.,  Google Scholar

[29]

D. Milder, The effects of truncation on surface-wave Hamiltonians, J. Fluid Mech., 217 (1990), 249-262.  doi: 10.1017/S0022112090000714.  Google Scholar

[30]

D. Nicholls, Spectral stability of traveling water waves: Eigenvalue collision, singularities, and direct numerical simulation, Phys. D, 240 (2011), 376-381, URL http://www.sciencedirect.com/science/article/pii/S0167278910002630. Google Scholar

[31]

P. Plotnikov and J. Toland, Modelling nonlinear hydroelastic waves, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 369 (2011), 2942-2956.  doi: 10.1098/rsta.2011.0104.  Google Scholar

[32]

M. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000.  doi: 10.1080/03605308208820242.  Google Scholar

[33]

M. Taylor, Pseudodifferential Operators and Nonlinear PDE, vol. 100 of Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1991. doi: 10.1007/978-1-4612-0431-2.  Google Scholar

[34]

J. Toland, Heavy hydroelastic travelling waves, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2371-2397.  doi: 10.1098/rspa.2007.1883.  Google Scholar

[35]

J. Wilkening and V. Vasan, Comparison of five methods of computing the Dirichlet-Neumann operator for the water wave problem, in Nonlinear wave equations: analytic and computational techniques, vol. 635 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2015, 175–210. doi: 10.1090/conm/635/12713.  Google Scholar

[36]

S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D, Invent. Math., 130 (1997), 39-72.  doi: 10.1007/s002220050177.  Google Scholar

[37]

S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445-495.  doi: 10.1090/S0894-0347-99-00290-8.  Google Scholar

[38]

V. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Journal of Applied Mechanics and Technical Physics, 9 (1968), 190-194.  doi: 10.1007/BF00913182.  Google Scholar

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