# American Institute of Mathematical Sciences

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

## Wave breaking of periodic solutions to the Fornberg-Whitham equation

 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.

Citation: Günther Hörmann. Wave breaking of periodic solutions to the Fornberg-Whitham equation. Discrete & Continuous Dynamical Systems - A, 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. [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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##### 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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