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. 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

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

[1]

Günther Hörmann, Hisashi Okamoto. Weak periodic solutions and numerical case studies of the Fornberg-Whitham equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4455-4469. doi: 10.3934/dcds.2019182

[2]

Min Li, Zhaoyang Yin. Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280

[3]

Frédéric Abergel, Jean-Michel Rakotoson. Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1809-1818. doi: 10.3934/dcds.2013.33.1809

[4]

Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103

[5]

Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831

[6]

Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108

[7]

Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1

[8]

Zhaoyang Yin. Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2004, 3 (3) : 501-508. doi: 10.3934/cpaa.2004.3.501

[9]

Min Zhu, Shuanghu Zhang. Blow-up of solutions to the periodic modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7235-7256. doi: 10.3934/dcds.2016115

[10]

Min Zhu, Ying Wang. Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 645-661. doi: 10.3934/dcds.2017027

[11]

Min Zhu, Shuanghu Zhang. On the blow-up of solutions to the periodic modified integrable Camassa--Holm equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2347-2364. doi: 10.3934/dcds.2016.36.2347

[12]

Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315

[13]

Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations & Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669

[14]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71

[15]

Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089

[16]

Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639

[17]

Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147

[18]

Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617

[19]

Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069

[20]

István Győri, Yukihiko Nakata, Gergely Röst. Unbounded and blow-up solutions for a delay logistic equation with positive feedback. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2845-2854. doi: 10.3934/cpaa.2018134

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (45)
  • HTML views (146)
  • Cited by (2)

Other articles
by authors

[Back to Top]