Steady water waves
HTML articles powered by AMS MathViewer
- by Walter A. Strauss PDF
- Bull. Amer. Math. Soc. 47 (2010), 671-694 Request permission
Erratum: Bull. Amer. Math. Soc. 48 (2011), 153-153.
Abstract:
We present a survey of certain aspects of the theory of steady water waves with emphasis on the role played by vorticity. Historical background, numerical illustrations, and brief discussions of the time-dependent problem and of approximate models are included as well.References
- G. B. Airy, Tides and waves, Encyclopedia Metropolitana 5 (1845), 241–396.
- Borys Alvarez-Samaniego and David Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math. 171 (2008), no. 3, 485–541. MR 2372806, DOI 10.1007/s00222-007-0088-4
- David M. Ambrose, Well-posedness of vortex sheets with surface tension, SIAM J. Math. Anal. 35 (2003), no. 1, 211–244. MR 2001473, DOI 10.1137/S0036141002403869
- David M. Ambrose and Nader Masmoudi, Well-posedness of 3D vortex sheets with surface tension, Commun. Math. Sci. 5 (2007), no. 2, 391–430. MR 2334849
- C. J. Amick and J. F. Toland, On solitary water-waves of finite amplitude, Arch. Rational Mech. Anal. 76 (1981), no. 1, 9–95. MR 629699, DOI 10.1007/BF00250799
- C. J. Amick and R. E. L. Turner, Small internal waves in two-fluid systems, Arch. Rational Mech. Anal. 108 (1989), no. 2, 111–139. MR 1011554, DOI 10.1007/BF01053459
- Charles J. Amick and Robert E. L. Turner, Center manifolds in equations from hydrodynamics, NoDEA Nonlinear Differential Equations Appl. 1 (1994), no. 1, 47–90. MR 1273343, DOI 10.1007/BF01194039
- J. Thomas Beale, The existence of solitary water waves, Comm. Pure Appl. Math. 30 (1977), no. 4, 373–389. MR 445136, DOI 10.1002/cpa.3160300402
- T. B. Benjamin and J. E. Feir, The disintegration of wavetrains in deep water, J. Fluid Mech. 27 (1967), 417–430.
- J. L. Bona, D. K. Bose, and R. E. L. Turner, Finite-amplitude steady waves in stratified fluids, J. Math. Pures Appl. (9) 62 (1983), no. 4, 389–439 (1984). MR 735931
- Jerry L. Bona, Thierry Colin, and David Lannes, Long wave approximations for water waves, Arch. Ration. Mech. Anal. 178 (2005), no. 3, 373–410. MR 2196497, DOI 10.1007/s00205-005-0378-1
- M. J. Boussinesq, Essai sur la théorie des eaux courantes, Mémoirs présentés par divers savants à l’Acad. des Sciences Inst. France (série 2) 23 (1877), 1–680.
- Thomas J. Bridges and Alexander Mielke, A proof of the Benjamin-Feir instability, Arch. Rational Mech. Anal. 133 (1995), no. 2, 145–198. MR 1367360, DOI 10.1007/BF00376815
- Boris Buffoni and John Toland, Analytic theory of global bifurcation, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2003. An introduction. MR 1956130, DOI 10.1515/9781400884339
- A. Cauchy, Théorie de la propagation des ondes à la surface d’un fluide pésant, Oeuvres Complètes 1 (1827).
- Adrian Constantin, On the deep water wave motion, J. Phys. A 34 (2001), no. 7, 1405–1417. MR 1819940, DOI 10.1088/0305-4470/34/7/313
- Adrian Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006), no. 3, 523–535. MR 2257390, DOI 10.1007/s00222-006-0002-5
- Adrian Constantin, Mats Ehrnström, and Erik Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J. 140 (2007), no. 3, 591–603. MR 2362244, DOI 10.1215/S0012-7094-07-14034-1
- Adrian Constantin and David Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal. 192 (2009), no. 1, 165–186. MR 2481064, DOI 10.1007/s00205-008-0128-2
- Adrian Constantin, David Sattinger, and Walter Strauss, Variational formulations for steady water waves with vorticity, J. Fluid Mech. 548 (2006), 151–163. MR 2264220, DOI 10.1017/S0022112005007469
- A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity (preprint 2010).
- Adrian Constantin and Walter Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math. 57 (2004), no. 4, 481–527. MR 2027299, DOI 10.1002/cpa.3046
- Adrian Constantin and Walter Strauss, Rotational steady water waves near stagnation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 365 (2007), no. 1858, 2227–2239. MR 2329144, DOI 10.1098/rsta.2007.2004
- Adrian Constantin and Walter A. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math. 60 (2007), no. 6, 911–950. MR 2306225, DOI 10.1002/cpa.20165
- Adrian Constantin and Walter Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math. 63 (2010), no. 4, 533–557. MR 2604871, DOI 10.1002/cpa.20299
- Daniel Coutand and Steve Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc. 20 (2007), no. 3, 829–930. MR 2291920, DOI 10.1090/S0894-0347-07-00556-5
- Walter Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Comm. Partial Differential Equations 10 (1985), no. 8, 787–1003. MR 795808, DOI 10.1080/03605308508820396
- Walter Craig and David P. Nicholls, Traveling gravity water waves in two and three dimensions, Eur. J. Mech. B Fluids 21 (2002), no. 6, 615–641. MR 1947187, DOI 10.1016/S0997-7546(02)01207-4
- Walter Craig and Peter Sternberg, Symmetry of solitary waves, Comm. Partial Differential Equations 13 (1988), no. 5, 603–633. MR 919444, DOI 10.1080/03605308808820554
- W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys. 108 (1993), no. 1, 73–83. MR 1239970, DOI 10.1006/jcph.1993.1164
- W. Craig, C. Sulem, and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity 5 (1992), no. 2, 497–522. MR 1158383
- V. Kreĭg and K. E. Veĭn, Mathematical aspects of surface waves on water, Uspekhi Mat. Nauk 62 (2007), no. 3(375), 95–116 (Russian, with Russian summary); English transl., Russian Math. Surveys 62 (2007), no. 3, 453–473. MR 2355420, DOI 10.1070/RM2007v062n03ABEH004413
- Alex D. D. Craik, The origins of water wave theory, Annual review of fluid mechanics. Vol. 36, Annu. Rev. Fluid Mech., vol. 36, Annual Reviews, Palo Alto, CA, 2004, pp. 1–28. MR 2062306, DOI 10.1146/annurev.fluid.36.050802.122118
- Michael G. Crandall and Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340. MR 0288640, DOI 10.1016/0022-1236(71)90015-2
- A. F. Teles 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. MR 985439, DOI 10.1017/S0022112088002423
- A. Dalmedico, La propagation de ondes en eau profonde et ses développements mathématiques, The History of Modern Mathematics, D. Rowe and J. McCleary, eds., 2 (1989), 129–168.
- E. N. Dancer, Global solution branches for positive mappings, Arch. Rational Mech. Anal. 52 (1973), 181–192. MR 353077, DOI 10.1007/BF00282326
- Olivier Darrigol, The spirited horse, the engineer, and the mathematician: water waves in nineteenth-century hydrodynamics, Arch. Hist. Exact Sci. 58 (2003), no. 1, 21–95. MR 2020055, DOI 10.1007/s00407-003-0070-5
- Olivier Darrigol, Worlds of flow, Oxford University Press, New York, 2005. A history of hydrodynamics from the Bernoullis to Prandtl. MR 2178164
- Frédéric Dias and Gérard Iooss, Water-waves as a spatial dynamical system, Handbook of mathematical fluid dynamics, Vol. II, North-Holland, Amsterdam, 2003, pp. 443–499. MR 1984157, DOI 10.1016/S1874-5792(03)80012-5
- M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d’ampleur finie, J. Math. Pures Appl. 13 (1934), 217–291.
- L. Euler, General principles of the motion of fluids, Hist. de l’Acad. de Berlin 11 (1755), 274–315.
- Charles L. Fefferman, Existence and smoothness of the Navier-Stokes equation, The millennium prize problems, Clay Math. Inst., Cambridge, MA, 2006, pp. 57–67. MR 2238274
- L. E. Fraenkel, An introduction to maximum principles and symmetry in elliptic problems, Cambridge Tracts in Mathematics, vol. 128, Cambridge University Press, Cambridge, 2000. MR 1751289, DOI 10.1017/CBO9780511569203
- K. O. Friedrichs, Uber ein minimumproblem für potential strömung mit freien rand, Math. Ann. 109 (1933), 60–82.
- P. Germain, Nader Masmoudi, and Jalal Shatah, Global solutions for the gravity water waves equation in dimension 3, C. R. Math. Acad. Sci. Paris 347 (2009), no. 15-16, 897–902 (English, with English and French summaries). MR 2542891, DOI 10.1016/j.crma.2009.05.005
- F. Gerstner, Theorie der wellen, Abhand. Koen. Boehmischen Gesel. Wiss., Prague, 1802.
- M. D. Groves, Steady water waves, J. Nonlinear Math. Phys. 11 (2004), no. 4, 435–460. MR 2097656, DOI 10.2991/jnmp.2004.11.4.2
- M. D. Groves and E. Wahlén, Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity, Phys. D 237 (2008), no. 10-12, 1530–1538. MR 2454604, DOI 10.1016/j.physd.2008.03.015
- Timothy J. Healey and Henry C. Simpson, Global continuation in nonlinear elasticity, Arch. Rational Mech. Anal. 143 (1998), no. 1, 1–28. MR 1643646, DOI 10.1007/s002050050098
- Vera Mikyoung Hur, Global bifurcation theory of deep-water waves with vorticity, SIAM J. Math. Anal. 37 (2006), no. 5, 1482–1521. MR 2215274, DOI 10.1137/040621168
- Vera Mikyoung Hur, Exact solitary water waves with vorticity, Arch. Ration. Mech. Anal. 188 (2008), no. 2, 213–244. MR 2385741, DOI 10.1007/s00205-007-0064-6
- —, Stokes waves with vorticity (preprint 2009).
- Gérard Iooss and Pavel I. Plotnikov, Small divisor problem in the theory of three-dimensional water gravity waves, Mem. Amer. Math. Soc. 200 (2009), no. 940, viii+128. MR 2529006, DOI 10.1090/memo/0940
- R. S. Johnson, A modern introduction to the mathematical theory of water waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997. MR 1629555, DOI 10.1017/CBO9780511624056
- Tadayoshi Kano and Takaaki Nishida, Sur les ondes de surface de l’eau avec une justification mathématique des équations des ondes en eau peu profonde, J. Math. Kyoto Univ. 19 (1979), no. 2, 335–370 (French). MR 545714, DOI 10.1215/kjm/1250522437
- G. Keady and J. Norbury, On the existence theory for irrotational water waves, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 1, 137–157. MR 502787, DOI 10.1017/S0305004100054372
- Sergiu Klainerman, Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math. 38 (1985), no. 5, 631–641. MR 803252, DOI 10.1002/cpa.3160380512
- Joy Ko and Walter Strauss, Large-amplitude steady rotational water waves, Eur. J. Mech. B Fluids 27 (2008), no. 2, 96–109. MR 2389493, DOI 10.1016/j.euromechflu.2007.04.004
- Joy Ko and Walter Strauss, Effect of vorticity on steady water waves, J. Fluid Mech. 608 (2008), 197–215. MR 2439751, DOI 10.1017/S0022112008002371
- D. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, Phil. Mag. 39 (1895), 422–443.
- Ju. P. Krasovskiĭ, On the theory of steady-state waves of finite amplitude, Ž. Vyčisl. Mat i Mat. Fiz. 1 (1961), 836–855 (Russian). MR 138284
- David Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc. 18 (2005), no. 3, 605–654. MR 2138139, DOI 10.1090/S0894-0347-05-00484-4
- Jean Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), no. 1, 193–248 (French). MR 1555394, DOI 10.1007/BF02547354
- T. Levi-Civita, Determinazione rigorosa delle onde irrotazionali periodiche in acqua profonda, Rend. Accad. Lincei 33 (1924), 141–150.
- Gary M. Lieberman and Neil S. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. Amer. Math. Soc. 295 (1986), no. 2, 509–546. MR 833695, DOI 10.1090/S0002-9947-1986-0833695-6
- Zhiwu Lin, On linear instability of 2D solitary water waves, Int. Math. Res. Not. IMRN 7 (2009), 1247–1303. MR 2495304, DOI 10.1093/imrn/rnn158
- Hans Lindblad, Well posedness for the motion of a compressible liquid with free surface boundary, Comm. Math. Phys. 260 (2005), no. 2, 319–392. MR 2177323, DOI 10.1007/s00220-005-1406-6
- Hans Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2) 162 (2005), no. 1, 109–194. MR 2178961, DOI 10.4007/annals.2005.162.109
- A. Nekrasov, On steady waves, Izv. Ivanovo-Voznesenk. Politekhn. 3 (1921).
- Hisashi Okamoto and Mayumi Sh\B{o}ji, The mathematical theory of permanent progressive water-waves, Advanced Series in Nonlinear Dynamics, vol. 20, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR 1869386, DOI 10.1142/4547
- Robert L. Pego and Michael I. Weinstein, Convective linear stability of solitary waves for Boussinesq equations, Stud. Appl. Math. 99 (1997), no. 4, 311–375. MR 1477120, DOI 10.1111/1467-9590.00063
- Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487–513. MR 0301587, DOI 10.1016/0022-1236(71)90030-9
- Guido Schneider and C. Eugene Wayne, The long-wave limit for the water wave problem. I. The case of zero surface tension, Comm. Pure Appl. Math. 53 (2000), no. 12, 1475–1535. MR 1780702, DOI 10.1002/1097-0312(200012)53:12<1475::AID-CPA1>3.0.CO;2-V
- Ben Schweizer, On the three-dimensional Euler equations with a free boundary subject to surface tension, Ann. Inst. H. Poincaré C Anal. Non Linéaire 22 (2005), no. 6, 753–781. MR 2172858, DOI 10.1016/j.anihpc.2004.11.001
- Jalal Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38 (1985), no. 5, 685–696. MR 803256, DOI 10.1002/cpa.3160380516
- Jalal Shatah and Chongchun Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation, Comm. Pure Appl. Math. 61 (2008), no. 5, 698–744. MR 2388661, DOI 10.1002/cpa.20213
- G. Stokes, On the theory of oscillatory waves, Trans. Camb. Phil. Soc. 8 (1847), 441–455.
- J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal. 7 (1996), no. 1, 1–48. MR 1422004, DOI 10.12775/TMNA.1996.001
- Eugen Varvaruca, On some properties of traveling water waves with vorticity, SIAM J. Math. Anal. 39 (2008), no. 5, 1686–1692. MR 2377294, DOI 10.1137/070697513
- Eugen Varvaruca, On the existence of extreme waves and the Stokes conjecture with vorticity, J. Differential Equations 246 (2009), no. 10, 4043–4076. MR 2514735, DOI 10.1016/j.jde.2008.12.018
- Erik Wahlén, A note on steady gravity waves with vorticity, Int. Math. Res. Not. 7 (2005), 389–396. MR 2130838, DOI 10.1155/IMRN.2005.389
- Erik Wahlén, Steady periodic capillary-gravity waves with vorticity, SIAM J. Math. Anal. 38 (2006), no. 3, 921–943. MR 2262949, DOI 10.1137/050630465
- S. Walsh, Steady periodic gravity waves with surface tension (preprint 2010).
- Samuel Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal. 41 (2009), no. 3, 1054–1105. MR 2529956, DOI 10.1137/080721583
- Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D, Invent. Math. 130 (1997), no. 1, 39–72. MR 1471885, DOI 10.1007/s002220050177
- Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc. 12 (1999), no. 2, 445–495. MR 1641609, DOI 10.1090/S0894-0347-99-00290-8
- Sijue Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math. 177 (2009), no. 1, 45–135. MR 2507638, DOI 10.1007/s00222-009-0176-8
- —, Global wellposedness of the $3$-$D$ full water wave problem (preprint 2009).
- V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 2 (1968), 190–194.
- Ping Zhang and Zhifei Zhang, On the free boundary problem of three-dimensional incompressible Euler equations, Comm. Pure Appl. Math. 61 (2008), no. 7, 877–940. MR 2410409, DOI 10.1002/cpa.20226
Additional Information
- Walter A. Strauss
- Affiliation: Department of Mathematics and Lefschetz Center for Dynamical Systems, Brown University, Providence, Rhode Island
- Email: wstrauss@math.brown.edu
- Received by editor(s): August 19, 2009
- Published electronically: July 20, 2010
- Additional Notes: I would like to thank my colleagues, Adrian Constantin, Joy Ko, and Samuel Walsh for their help with this paper, as well as the anonymous referees.
The work of this author was supported in part by NSF Grant DMS-0405066. - © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Bull. Amer. Math. Soc. 47 (2010), 671-694
- MSC (2010): Primary 76B15, 35Q31, 35R35
- DOI: https://doi.org/10.1090/S0273-0979-2010-01302-1
- MathSciNet review: 2721042