-
Previous Article
On substitution tilings and Delone sets without finite local complexity
- DCDS Home
- This Issue
-
Next Article
Classification of traveling waves for a quadratic Szegő equation
Sufficiently strong dispersion removes ill-posedness in truncated series models of water waves
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 |
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.
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. |
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. |
[1] |
David M. Ambrose, Jerry L. Bona, David P. Nicholls. Well-posedness of a model for water waves with viscosity. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1113-1137. doi: 10.3934/dcdsb.2012.17.1113 |
[2] |
Zhaoyang Yin. Well-posedness, blowup, and global existence for an integrable shallow water equation. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 393-411. doi: 10.3934/dcds.2004.11.393 |
[3] |
Pavel Krejčí, Elisabetta Rocca. Well-posedness of an extended model for water-ice phase transitions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 439-460. doi: 10.3934/dcdss.2013.6.439 |
[4] |
Bashar Khorbatly. Long, intermediate and short-term well-posedness of high precision shallow-water models with topography variations. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022068 |
[5] |
K. Domelevo. Well-posedness of a kinetic model of dispersed two-phase flow with point-particles and stability of travelling waves. Discrete and Continuous Dynamical Systems - B, 2002, 2 (4) : 591-607. doi: 10.3934/dcdsb.2002.2.591 |
[6] |
Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093 |
[7] |
Ademir Pastor. On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2217-2242. doi: 10.3934/cpaa.2019100 |
[8] |
Jaime Angulo, Carlos Matheus, Didier Pilod. Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system. Communications on Pure and Applied Analysis, 2009, 8 (3) : 815-844. doi: 10.3934/cpaa.2009.8.815 |
[9] |
Vanessa Barros, Felipe Linares. A remark on the well-posedness of a degenerated Zakharov system. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1259-1274. doi: 10.3934/cpaa.2015.14.1259 |
[10] |
Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007 |
[11] |
Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036 |
[12] |
Elissar Nasreddine. Well-posedness for a model of individual clustering. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2647-2668. doi: 10.3934/dcdsb.2013.18.2647 |
[13] |
Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241 |
[14] |
Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1 |
[15] |
A. Alexandrou Himonas, Curtis Holliman. On well-posedness of the Degasperis-Procesi equation. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 469-488. doi: 10.3934/dcds.2011.31.469 |
[16] |
Mircea Sofonea, Yi-bin Xiao. Tykhonov well-posedness of a viscoplastic contact problem†. Evolution Equations and Control Theory, 2020, 9 (4) : 1167-1185. doi: 10.3934/eect.2020048 |
[17] |
Nils Strunk. Well-posedness for the supercritical gKdV equation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 527-542. doi: 10.3934/cpaa.2014.13.527 |
[18] |
Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations and Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15 |
[19] |
Hongjun Gao, Chengfeng Sun. Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3053-3073. doi: 10.3934/dcdsb.2016087 |
[20] |
Hongwei Wang, Amin Esfahani. Well-posedness and asymptotic behavior of the dissipative Ostrovsky equation. Evolution Equations and Control Theory, 2019, 8 (4) : 709-735. doi: 10.3934/eect.2019035 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]