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Traveling wave solutions of a highly nonlinear shallow water equation
Wave breaking of periodic solutions to the Fornberg-Whitham equation
Fakultät für Mathematik, Universität Wien, Austria |
Based on recent well-posedness results in Sobolev (or Besov spaces) for periodic solutions to the Fornberg-Whitham equations we investigate here the questions of wave breaking and blow-up for these solutions. We show first that finite maximal life time of a solution necessarily leads to wave breaking. Second, we prove that for a certain class of initial wave profiles the corresponding solutions do indeed blow-up in finite time.
References:
| [1] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011. |
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A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2011. |
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A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
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A. Constantin, J. Escher, R. S. Johnson and G. Villari, Nonlinear Water Waves, Springer-Verlag, Florence, 2016. |
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B. Fornberg and G. B. Whitham,
A numerical and theoretical study of certain nonlinear wave phenomena, Philos. Trans. Roy. Soc. London Ser. A, 289 (1978), 373-404.
doi: 10.1098/rsta.1978.0064. |
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L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Upper Saddle River, 2004. |
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J. Holmes,
Well-posedness of the Fornberg-Whitham equation on the circle, J. Differential Equations, 260 (2016), 8530-8549.
doi: 10.1016/j.jde.2016.02.030. |
| [8] |
J. Holmes and R. C. Thompson,
Well-posedness and continuity properties of the Fornberg-Whitham equation in Besov spaces, J. Differential Equations, 263 (2017), 4355-4381.
doi: 10.1016/j.jde.2017.05.019. |
| [9] |
P. I. Naumkin and I. A. Shishmarëv, Nonlinear Nonlocal Equations in the Theory of Waves, American Mathematical Society, Providence, 1994. |
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R. L. Seliger,
A note on the breaking of waves, Proc. Roy. Soc. A, 303 (1968), 493-496.
doi: 10.1098/rspa.1968.0063. |
| [11] |
G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York-London-Sydney, 1974. |
show all references
References:
| [1] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011. |
| [2] |
A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2011. |
| [3] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
| [4] |
A. Constantin, J. Escher, R. S. Johnson and G. Villari, Nonlinear Water Waves, Springer-Verlag, Florence, 2016. |
| [5] |
B. Fornberg and G. B. Whitham,
A numerical and theoretical study of certain nonlinear wave phenomena, Philos. Trans. Roy. Soc. London Ser. A, 289 (1978), 373-404.
doi: 10.1098/rsta.1978.0064. |
| [6] |
L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Upper Saddle River, 2004. |
| [7] |
J. Holmes,
Well-posedness of the Fornberg-Whitham equation on the circle, J. Differential Equations, 260 (2016), 8530-8549.
doi: 10.1016/j.jde.2016.02.030. |
| [8] |
J. Holmes and R. C. Thompson,
Well-posedness and continuity properties of the Fornberg-Whitham equation in Besov spaces, J. Differential Equations, 263 (2017), 4355-4381.
doi: 10.1016/j.jde.2017.05.019. |
| [9] |
P. I. Naumkin and I. A. Shishmarëv, Nonlinear Nonlocal Equations in the Theory of Waves, American Mathematical Society, Providence, 1994. |
| [10] |
R. L. Seliger,
A note on the breaking of waves, Proc. Roy. Soc. A, 303 (1968), 493-496.
doi: 10.1098/rspa.1968.0063. |
| [11] |
G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York-London-Sydney, 1974. |
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