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

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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 and Continuous Dynamical Systems, 2019, 39 (6) : 3123-3147. doi: 10.3934/dcds.2019129

References:

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[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.
[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.
[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.
[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.
[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.
[7]	D. Ambrose and M. Siegel, Ill-posedness of quadratic truncated series models of gravity water waves, Preprint.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83. doi: 10.1006/jcph.1993.1164.
[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.
[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.
[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.
[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.
[22]	L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, vol. 26 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 1997.
[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.
[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.
[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.
[26]	S. Liu, Well-posedness of Hydroelastic Waves and Their Truncated Series Models, PhD thesis, Drexel University, 2016.
[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.
[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.,
[29]	D. Milder, The effects of truncation on surface-wave Hamiltonians, J. Fluid Mech., 217 (1990), 249-262. doi: 10.1017/S0022112090000714.
[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.
[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.
[32]	M. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000. doi: 10.1080/03605308208820242.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[7]	D. Ambrose and M. Siegel, Ill-posedness of quadratic truncated series models of gravity water waves, Preprint.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83. doi: 10.1006/jcph.1993.1164.
[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.
[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.
[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.
[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.
[22]	L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, vol. 26 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 1997.
[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.
[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.
[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.
[26]	S. Liu, Well-posedness of Hydroelastic Waves and Their Truncated Series Models, PhD thesis, Drexel University, 2016.
[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.
[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.,
[29]	D. Milder, The effects of truncation on surface-wave Hamiltonians, J. Fluid Mech., 217 (1990), 249-262. doi: 10.1017/S0022112090000714.
[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.
[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.
[32]	M. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), 959-1000. doi: 10.1080/03605308208820242.
[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.
[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.
[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.
[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.
[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.
[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.

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