Large-Amplitude Solitary Water Waves with Vorticity
Abstract
We provide the first construction of exact solitary waves of large amplitude with an arbitrary distribution of vorticity. We use continuation to construct a global connected set of symmetric solitary waves of elevation, whose profiles decrease monotonically on either side of a central crest. This generalizes the classical result of Amick and Toland.
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References
1.
S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), pp. 623--727.
2.
C. J. Amick and J. F. Toland, On periodic water-waves and their convergence to solitary waves in the long-wave limit, Philos. Trans. Roy. Soc. London Ser. A, 303 (1981), pp. 633--669.
3.
C. J. Amick and J. F. Toland, On solitary water-waves of finite amplitude, Arch. Ration. Mech. Anal., 76 (1981), pp. 9--95.
4.
J. T. Beale, The existence of solitary water waves, Comm. Pure Appl. Math., 30 (1977), pp. 373--389.
5.
T. B. Benjamin, The solitary wave on a stream with an arbitrary distribution of vorticity, J. Fluid Mech., 12 (1962), pp. 97--116.
6.
T. B. Benjamin, J. L. Bona, and D. K. Bose, Solitary-wave solutions of nonlinear problems, Philos. Trans. Roy. Soc. London Ser. A, 331 (1990), pp. 195--244.
7.
J. C. Burns, Long waves in running water, Math. Proc. Cambridge Philos. Soc., 49 (1953), pp. 695--706.
8.
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.
9.
A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), pp. 481--527.
10.
A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Ration. Mech. Anal., 202 (2011), pp. 133--175.
11.
W. Craig and P. Sternberg, Symmetry of solitary waves, Comm. Partial Differential Equations, 13 (1988), pp. 603--633.
12.
M. L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'amplitude finie, Journ. de Math., 13 (1934), pp. 217--289.
13.
P. M. Fitzpatrick, J. Pejsachowicz, and P. J. Rabier, The degree of proper $C^2$ Fredholm mappings. I, J. Reine Angew. Math., 427 (1992), pp. 1--33.
14.
N. C. Freeman and R. S. Johnson, Shallow water waves on shear flows, J. Fluid Mech., 42 (1970), pp. 401--409.
15.
K. O. Friedrichs and D. H. Hyers, The existence of solitary waves, Comm. Pure Appl. Math., 7 (1954), pp. 517--550.
16.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
17.
M. D. Groves and E. Wahlén, Spatial dynamics methods for solitary gravity-capillary water waves with an arbitrary distribution of vorticity, SIAM J. Math. Anal., 39 (2007), pp. 932--964.
18.
M. D. Groves and E. Wahlén, Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity, Phys. D, 237 (2008), pp. 1530--1538.
19.
T. J. Healey and H. C. Simpson, Global continuation in nonlinear elasticity, Arch. Ration. Mech. Anal., 143 (1998), pp. 1--28.
20.
V. M. Hur, Exact solitary water waves with vorticity, Arch. Ration. Mech. Anal., 188 (2008), pp. 213--244.
21.
V. M. Hur, Symmetry of solitary water waves with vorticity, Math. Res. Lett., 15 (2008), pp. 491--509.
22.
V. M. Hur, Analyticity of rotational flows beneath solitary water waves, Internat. Math. Research Notices, 2012 (2011), pp. 2550--2570.
23.
T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Grundlehren Math. Wiss. 132, Springer-Verlag, Berlin, 1976.
24.
G. Keady and J. Norbury, On the existence theory for irrotational water waves, Math. Proc. Cambridge Philos. Soc., 83 (1978), pp. 137--157.
25.
G. Keady and J. Norbury, Waves and conjugate streams with vorticity, Mathematika, 25 (1978), pp. 129--150.
26.
H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Appl. Math. Sci. 156, Springer-Verlag, New York, 2004.
27.
J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), pp. 197--215.
28.
V. Kozlov and N. Kuznetsov, On behaviour of free-surface profiles for bounded steady water waves, J. Math. Pures Appl. (9), 90 (2008), pp. 1--14.
29.
J. P. Krasovskiĭ, On the theory of steady-state waves of finite amplitude, Ž. Vyčisl. Mat. i Mat. Fiz., 1 (1961), pp. 836--855.
30.
M. A. Lavrentiev, I. On the theory of long waves. II. A contribution to the theory of long waves, Amer. Math. Soc. Transl., 1954 (1954), p. 53.
31.
T. Levi-Civita, Determinazione rigorosa delle onde irrotazionali periodiche in acqua profonda, Rend. Accad. Lincei, 33 (1924), pp. 141--150.
32.
G. M. Lieberman, Two-dimensional nonlinear boundary value problems for elliptic equations, Trans. Amer. Math. Soc., 300 (1987), pp. 287--295.
33.
J. B. McLeod, The Froude number for solitary waves, Proc. Roy. Soc. Edinburgh Sect. A, 97 (1984), pp. 193--197.
34.
J. B. McLeod, The Stokes and Krasovskii conjectures for the wave of greatest height, Stud. Appl. Math., 98 (1997), pp. 311--333.
35.
A. Mielke, Über maximalel p-regularität für differentialgleichungen in Banach-und Hilbert-Räumen, Math. Ann., 277 (1987), pp. 121--133.
36.
A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci., 10 (1988), pp. 51--66.
37.
A. I. Nekrasov, On steady waves, Izv. Ivanovo-Voznesensk. Politekhn. In-ta, 3 (1921).
38.
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1967.
39.
P. J. Rabier, Ascoli's theorem for functions vanishing at infinity and selected applications, J. Math. Anal. Appl., 290 (2004), pp. 171--189.
40.
P. J. Rabier and C. A. Stuart, Fredholm and properness properties of quasilinear elliptic operators on $\bold R^N$, Math. Nachr., 231 (2001), pp. 129--168.
41.
P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), pp. 487--513.
42.
G. G. Stokes, On the theory of oscillatory waves, Trans. Cambridge Philos. Soc., 8 (1847), pp. 441--473.
43.
A. M. Ter-Krikorov, The existence of periodic waves which degenerate into a solitary wave, J. Appl. Math. Mech., 24 (1960), pp. 930--949.
44.
A. M. Ter-Krikorov, The solitary wave on the surface of a turbulent liquid, Zh. Vychisl. Mat. Mat. Fiz., 1 (1961), pp. 1077--1088.
45.
J. F. Toland, On the existence of a wave of greatest height and Stokes's conjecture, Proc. Roy. Soc. London Ser. A, 363 (1978), pp. 469--485.
46.
E. Varvaruca, On the existence of extreme waves and the Stokes conjecture with vorticity, J. Differential Equations, 246 (2009), pp. 4043--4076.
47.
V. Volpert, Elliptic Partial Differential Equations: Volume 1: Fredholm Theory of Elliptic Problems in Unbounded Domains, Monogr. Math. 101, Birkhäuser, Basel, 2011.
48.
V. Volpert and A. Volpert, Properness and topological degree for general elliptic operators, Abstr. Appl. Anal., 2003 (2003), pp. 129--181.
Information & Authors
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Published In
SIAM Journal on Mathematical Analysis
Pages: 2937 - 2994
DOI: 10.1137/120891460
ISSN (online): 1095-7154
Copyright
© 2013, Society for Industrial and Applied Mathematics.
History
Submitted: 14 September 2012
Accepted: 24 June 2013
Published online: 26 September 2013
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