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March  2018, 38(3): 1605-1613. doi: 10.3934/dcds.2018066

Wave breaking of periodic solutions to the Fornberg-Whitham equation

Günther Hörmann

Fakultät für Mathematik, Universität Wien, Austria

Received  September 2017 Revised  October 2017 Published  December 2017

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.

Keywords: Periodic Fornberg-Whitham equation, wave breaking, blow-up, Sobolev spaces.
Mathematics Subject Classification: Primary: 35Q53, 35B44; Secondary: 35B10.
Citation: Günther Hörmann. Wave breaking of periodic solutions to the Fornberg-Whitham equation. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1605-1613. doi: 10.3934/dcds.2018066
References:
[1]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.  Google Scholar

[2]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2011.  Google Scholar

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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.  Google Scholar

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A. Constantin, J. Escher, R. S. Johnson and G. Villari, Nonlinear Water Waves, Springer-Verlag, Florence, 2016.  Google Scholar

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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.  Google Scholar

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L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Upper Saddle River, 2004.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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P. I. Naumkin and I. A. Shishmarëv, Nonlinear Nonlocal Equations in the Theory of Waves, American Mathematical Society, Providence, 1994.  Google Scholar

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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.  Google Scholar

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G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York-London-Sydney, 1974.  Google Scholar

show all references

References:
[1]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.  Google Scholar

[2]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2011.  Google Scholar

[3]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[4]

A. Constantin, J. Escher, R. S. Johnson and G. Villari, Nonlinear Water Waves, Springer-Verlag, Florence, 2016.  Google Scholar

[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.  Google Scholar

[6]

L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Upper Saddle River, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

P. I. Naumkin and I. A. Shishmarëv, Nonlinear Nonlocal Equations in the Theory of Waves, American Mathematical Society, Providence, 1994.  Google Scholar

[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.  Google Scholar

[11]

G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York-London-Sydney, 1974.  Google Scholar

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