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On the formation of singularities for surface water waves
Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity
1. | Institute of Mathematics Simion Stoilow of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest |
References:
[1] |
B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541.
doi: 10.1007/s00222-007-0088-4. |
[2] |
P. F. Byrd and M. D. Friedman, "Handbook of Elliptic Integrals for Engineers and Scientists," Springer-Verlag, Berlin, Heidelberg, New York, 1971. |
[3] |
A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417.
doi: 10.1088/0305-4470/34/7/313. |
[4] |
A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[5] |
A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B Fluids, 30 (2011), 12-16.
doi: 10.1016/j.euromechflu.2010.09.008. |
[6] |
A. Constantin, M. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves, Nonlinear Anal. Real World Appl., 9 (2008), 1336-1344.
doi: 10.1016/j.nonrwa.2007.03.003. |
[7] |
A. Constantin, M. Ehrnström M. and E. Wahlen, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603.
doi: 10.1215/S0012-7094-07-14034-1. |
[8] |
A. Constantin and J. Escher, Symmetry of steady periodic water waves with vorticity, J. Fluid. Mech., 498 (2004), 171-181.
doi: 10.1017/S0022112003006773. |
[9] |
A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity, Eur. J. Appl. Math., 15 (2004), 755-768.
doi: 10.1017/S0956792504005777. |
[10] |
A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.
doi: 10.1090/S0273-0979-07-01159-7. |
[11] |
A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568.
doi: 10.4007/annals.2011.173.1.12. |
[12] |
A. Constantin and G. Villari, Particle trajectories in linear water waves, J. Math. Fluid Mech., 10 (2008), 1-18.
doi: 10.1007/s00021-005-0214-2. |
[13] |
A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527.
doi: 10.1002/cpa.3046. |
[14] |
A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557.
doi: 10.1002/cpa.20299. |
[15] |
A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Arch. Ration. Mech. Anal., 199 (2011), 33-67.
doi: 10.1007/s00205-010-0314-x. |
[16] |
T. A. Da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid. Mech., 195 (1988), 281-302.
doi: 10.1017/S0022112088002423. |
[17] |
L. Debnath, "Nonlinear Water Waves," Boston, MA: Academic Press Inc., 1994. |
[18] |
M.-L. Dubreil-Jacotin, Sur la determination rigoureuse des ondes permanentes periodiques dempleur finie, J. Math. Pures Appl., 13 (1934), 217-291. |
[19] |
M. Ehrnström and G. Villari, Linear water waves with vorticity: Rotational features and particle paths, J. Differential Equations, 244 (2008), 1888-1909.
doi: 10.1016/j.jde.2008.01.012. |
[20] |
F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys. 2 (1809), 412-445.
doi: 10.1002/andp.18090320808. |
[21] |
D. Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not. (2006), Art. ID 23405, 13 pp.
doi: 10.1155/IMRN/2006/23405. |
[22] |
D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves, J. Nonlinear Math. Phys., 14 (2007), 1-7.
doi: 10.2991/jnmp.2007.14.1.1. |
[23] |
D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity water waves, Phil. Trans. R. Soc. A, 365 (2007), 2241-2251.
doi: 10.1098/rsta.2007.2005. |
[24] |
D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95.
doi: 10.2991/jnmp.2008.15.s2.7. |
[25] |
V. M. Hur, Global bifurcation theory of deep-water waves with vorticity, SIAM J. Math Anal., 37 (2006), 1482-1521.
doi: 10.1137/040621168. |
[26] |
V. M. Hur, Symmetry of steady periodic water waves with vorticity, Phil. Trans. R. Soc. A, 365 (2007), 2203-2214.
doi: 10.1098/rsta.2007.2002. |
[27] |
D. Ionescu-Kruse, Particle trajectories in linearized irrotational shallow water flows, J. Nonlinear Math. Phys., 15 (2008), 13-27.
doi: 10.2991/jnmp.2008.15.s2.2. |
[28] |
D. Ionescu-Kruse, Particle trajectories beneath small amplitude shallow water waves in constant vorticity flows, Nonlinear Anal-Theor, 71 (2009), 3779-3793.
doi: 10.1016/j.na.2009.02.050. |
[29] |
D. Ionescu-Kruse, Exact solutions for small-amplitude capillary-gravity water waves, Wave Motion, 46 (2009), 379-388.
doi: 10.1016/j.wavemoti.2009.06.003. |
[30] |
D. Ionescu-Kruse, Small-amplitude capillary-gravity water waves: exact solutions and particle motion beneath such waves, Nonlinear Anal. Real World Appl., 11 (2010), 2989-3000.
doi: 10.1016/j.nonrwa.2009.10.019. |
[31] |
D. Ionescu-Kruse, Peakons arising as particl epaths beneath small-amplitude water waves in cosntant vorticity flows, J. Nonlinear Math. Phys., 17 (2010), 415-422.
doi: 10.1142/S140292511000101X. |
[32] |
R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves," Cambridge Univeristy Press, 1997.
doi: 10.1017/CBO9780511624056. |
[33] |
H. Kalisch, Periodic traveling water waves with isobaric streamlines, J. Nonlinear Math. Phys., 11 (2004), 461-471.
doi: 10.2991/jnmp.2004.11.4.3. |
[34] |
H. Lamb, "Hydrodynamics," 6th ed., Cambridge University Press, 1953. |
[35] |
J. Lighthill, "Waves in Fluids," Cambridge University Press, 2001. |
[36] |
A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts Appl. Math., vol. 27, Cambridge University Press, 2002. |
[37] |
W. J. M. Rankine, On the exact form of waves near the surface of deep water, Phil. Trans. R. Soc. A, 153 (1863), 127-138.
doi: 10.1098/rstl.1863.0006. |
[38] |
V. Smirnov, "Cours de Mathématiques supérieures, Tome III, deuxième partie," Mir, Moscou, 1972. |
[39] |
W. Strauss, Steady water waves, Bull. Amer. Math. Soc., 47 (2010), 671-694.
doi: 10.1090/S0273-0979-2010-01302-1. |
[40] |
E. Varvaruca, On the existence of extreme waves and the Stokes conjecture with vorticity, J. Differential Equations, 246 (2009), 4043-4076.
doi: 10.1016/j.jde.2008.12.018. |
[41] |
E. Wahlen, On rotational water waves with surface tension, Phil. Trans. R. Soc. A, 365 (2007), 2215-2225.
doi: 10.1098/rsta.2007.2003. |
[42] |
E. Wahlen, Steady water waves with a critical layer, J. Differential Eq., 246 (2009), 2468-2483.
doi: 10.1016/j.jde.2008.10.005. |
show all references
References:
[1] |
B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541.
doi: 10.1007/s00222-007-0088-4. |
[2] |
P. F. Byrd and M. D. Friedman, "Handbook of Elliptic Integrals for Engineers and Scientists," Springer-Verlag, Berlin, Heidelberg, New York, 1971. |
[3] |
A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417.
doi: 10.1088/0305-4470/34/7/313. |
[4] |
A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[5] |
A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B Fluids, 30 (2011), 12-16.
doi: 10.1016/j.euromechflu.2010.09.008. |
[6] |
A. Constantin, M. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves, Nonlinear Anal. Real World Appl., 9 (2008), 1336-1344.
doi: 10.1016/j.nonrwa.2007.03.003. |
[7] |
A. Constantin, M. Ehrnström M. and E. Wahlen, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603.
doi: 10.1215/S0012-7094-07-14034-1. |
[8] |
A. Constantin and J. Escher, Symmetry of steady periodic water waves with vorticity, J. Fluid. Mech., 498 (2004), 171-181.
doi: 10.1017/S0022112003006773. |
[9] |
A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity, Eur. J. Appl. Math., 15 (2004), 755-768.
doi: 10.1017/S0956792504005777. |
[10] |
A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.
doi: 10.1090/S0273-0979-07-01159-7. |
[11] |
A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568.
doi: 10.4007/annals.2011.173.1.12. |
[12] |
A. Constantin and G. Villari, Particle trajectories in linear water waves, J. Math. Fluid Mech., 10 (2008), 1-18.
doi: 10.1007/s00021-005-0214-2. |
[13] |
A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527.
doi: 10.1002/cpa.3046. |
[14] |
A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557.
doi: 10.1002/cpa.20299. |
[15] |
A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Arch. Ration. Mech. Anal., 199 (2011), 33-67.
doi: 10.1007/s00205-010-0314-x. |
[16] |
T. A. Da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid. Mech., 195 (1988), 281-302.
doi: 10.1017/S0022112088002423. |
[17] |
L. Debnath, "Nonlinear Water Waves," Boston, MA: Academic Press Inc., 1994. |
[18] |
M.-L. Dubreil-Jacotin, Sur la determination rigoureuse des ondes permanentes periodiques dempleur finie, J. Math. Pures Appl., 13 (1934), 217-291. |
[19] |
M. Ehrnström and G. Villari, Linear water waves with vorticity: Rotational features and particle paths, J. Differential Equations, 244 (2008), 1888-1909.
doi: 10.1016/j.jde.2008.01.012. |
[20] |
F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys. 2 (1809), 412-445.
doi: 10.1002/andp.18090320808. |
[21] |
D. Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not. (2006), Art. ID 23405, 13 pp.
doi: 10.1155/IMRN/2006/23405. |
[22] |
D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves, J. Nonlinear Math. Phys., 14 (2007), 1-7.
doi: 10.2991/jnmp.2007.14.1.1. |
[23] |
D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity water waves, Phil. Trans. R. Soc. A, 365 (2007), 2241-2251.
doi: 10.1098/rsta.2007.2005. |
[24] |
D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95.
doi: 10.2991/jnmp.2008.15.s2.7. |
[25] |
V. M. Hur, Global bifurcation theory of deep-water waves with vorticity, SIAM J. Math Anal., 37 (2006), 1482-1521.
doi: 10.1137/040621168. |
[26] |
V. M. Hur, Symmetry of steady periodic water waves with vorticity, Phil. Trans. R. Soc. A, 365 (2007), 2203-2214.
doi: 10.1098/rsta.2007.2002. |
[27] |
D. Ionescu-Kruse, Particle trajectories in linearized irrotational shallow water flows, J. Nonlinear Math. Phys., 15 (2008), 13-27.
doi: 10.2991/jnmp.2008.15.s2.2. |
[28] |
D. Ionescu-Kruse, Particle trajectories beneath small amplitude shallow water waves in constant vorticity flows, Nonlinear Anal-Theor, 71 (2009), 3779-3793.
doi: 10.1016/j.na.2009.02.050. |
[29] |
D. Ionescu-Kruse, Exact solutions for small-amplitude capillary-gravity water waves, Wave Motion, 46 (2009), 379-388.
doi: 10.1016/j.wavemoti.2009.06.003. |
[30] |
D. Ionescu-Kruse, Small-amplitude capillary-gravity water waves: exact solutions and particle motion beneath such waves, Nonlinear Anal. Real World Appl., 11 (2010), 2989-3000.
doi: 10.1016/j.nonrwa.2009.10.019. |
[31] |
D. Ionescu-Kruse, Peakons arising as particl epaths beneath small-amplitude water waves in cosntant vorticity flows, J. Nonlinear Math. Phys., 17 (2010), 415-422.
doi: 10.1142/S140292511000101X. |
[32] |
R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves," Cambridge Univeristy Press, 1997.
doi: 10.1017/CBO9780511624056. |
[33] |
H. Kalisch, Periodic traveling water waves with isobaric streamlines, J. Nonlinear Math. Phys., 11 (2004), 461-471.
doi: 10.2991/jnmp.2004.11.4.3. |
[34] |
H. Lamb, "Hydrodynamics," 6th ed., Cambridge University Press, 1953. |
[35] |
J. Lighthill, "Waves in Fluids," Cambridge University Press, 2001. |
[36] |
A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts Appl. Math., vol. 27, Cambridge University Press, 2002. |
[37] |
W. J. M. Rankine, On the exact form of waves near the surface of deep water, Phil. Trans. R. Soc. A, 153 (1863), 127-138.
doi: 10.1098/rstl.1863.0006. |
[38] |
V. Smirnov, "Cours de Mathématiques supérieures, Tome III, deuxième partie," Mir, Moscou, 1972. |
[39] |
W. Strauss, Steady water waves, Bull. Amer. Math. Soc., 47 (2010), 671-694.
doi: 10.1090/S0273-0979-2010-01302-1. |
[40] |
E. Varvaruca, On the existence of extreme waves and the Stokes conjecture with vorticity, J. Differential Equations, 246 (2009), 4043-4076.
doi: 10.1016/j.jde.2008.12.018. |
[41] |
E. Wahlen, On rotational water waves with surface tension, Phil. Trans. R. Soc. A, 365 (2007), 2215-2225.
doi: 10.1098/rsta.2007.2003. |
[42] |
E. Wahlen, Steady water waves with a critical layer, J. Differential Eq., 246 (2009), 2468-2483.
doi: 10.1016/j.jde.2008.10.005. |
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