Notes on wave-breaking phenomena for a Fornberg-Whitham-type equation. (English) Zbl 1514.35367
Summary: We investigate the wave-breaking mechanism of solutions to a Fornberg-Whitham-type equation. By means of a weaker conserved \(L^2\)-norm, we show some wave-breaking criteria for the equation, which improve some blow-up results on the classical Fornberg-Whitham equation. Moreover, our conclusions suggest that the wave breaking for the equation may occur even with small slope of the initial value. To analyze the interaction between nonlinear and nonlocal dispersion terms, we provide two different approaches. The first one is based on a subtle analysis on evolution of the solution \(u\) and its gradient \(u_x\). The other is to make full use of the known blow-up results on Riccati-type inequalities with \(t\)-dependent functions.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35B44 | Blow-up in context of PDEs |
| 76B25 | Solitary waves for incompressible inviscid fluids |
| 35D35 | Strong solutions to PDEs |
References:
| [1] | Constantin, A., Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50, 321-362 (2000) · Zbl 0944.35062 |
| [2] | Constantin, A.; Escher, J., On the structure of a family of quasilinear equations arising in shallow water theory, Math. Ann., 312, 403-416 (1998) · Zbl 0923.76028 |
| [3] | Constantin, A.; Escher, J., Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181, 229-243 (1998) · Zbl 0923.76025 |
| [4] | Constantin, A.; Lannes, D., The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192, 165-186 (2009) · Zbl 1169.76010 |
| [5] | Degasperis, A.; Procesi, M., Asymptotic integrability, (Degasperis, A.; Gaeta, G., Symmetry and Perturbation Theory (1999), World Scientific: World Scientific Singapore), 23-37 · Zbl 0963.35167 |
| [6] | Escher, J.; Liu, Y.; Yin, Z., Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation, Indiana Univ. Math. J., 56, 87-117 (2007) · Zbl 1124.35041 |
| [7] | Fornberg, G.; Whitham, G. B., A numerical and theoretical study of certain nonlinear wave phenomena, Philos. Trans. R. Soc. Lond. Ser. A, 289, 1361, 373-404 (1978) · Zbl 0384.65049 |
| [8] | Fu, Y.; Liu, Y.; Qu, C. Z., On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations, J. Funct. Anal., 262, 3125-3158 (2012) · Zbl 1234.35222 |
| [9] | Haziot, S., Wave breaking for the Fornberg-Whitham equation, J. Differ. Equ., 263, 8178-8185 (2017) · Zbl 1375.35375 |
| [10] | Holmes, J., Well-posedness of the Fornberg-Whitham equation on the circle, J. Differ. Equ., 260, 12, 8530-8549 (2016) · Zbl 1339.35268 |
| [11] | Holmes, J.; Thompson, R. C., Well-posedness and continuity properties of the Fornberg-Whitham equation in Besov spaces, J. Differ. Equ., 263, 7, 4355-4381 (2017) · Zbl 1372.35076 |
| [12] | Hörmann, G., Wave breaking of periodic solutions to the Fornberg-Whitham equation, Discrete Contin. Dyn. Syst., 38, 3, 1605-1613 (2018) · Zbl 1397.35254 |
| [13] | Hörmann, G., Discontinuous traveling waves as weak solutions to the Fornberg-Whitham equation, J. Differ. Equ., 265, 7, 2825-2841 (2018) · Zbl 1394.35099 |
| [14] | Hörmann, G., Solution concepts, well-posedness, and wave breaking for the Fornberg-Whitham equation, Monatshefte Math., 195, 421-449 (2021) · Zbl 1467.35002 |
| [15] | Hörmann, G.; Okamoto, H., Weak periodic solutions and numerical case studies of the Fornberg-Whitham equation, Discrete Contin. Dyn. Syst., 39, 8, 4455-4469 (2019) · Zbl 1415.35242 |
| [16] | Kato, T., Quasi-linear equations of evolution with applications to partial differential equations, (Spectral Theory and Differential Equations. Spectral Theory and Differential Equations, Lecture Notes in Math., vol. 448 (1975), Springer-Verlag: Springer-Verlag Berlin), 25-70 · Zbl 0315.35077 |
| [17] | Korteweg, D. J.; de Vries, G., On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves, Philos. Mag., 39, 5, 422-442 (1895) · JFM 26.0881.02 |
| [18] | Lai, S. Y.; Luo, K. X., Wave breaking to a shallow water wave equation involving the Fornberg-Whitham model, J. Differ. Equ., 344, 509-521 (2023) · Zbl 1503.35169 |
| [19] | Li, N.; Lai, S. Y., The entropy weak solution to a generalized Forberg-Whitham equation, Bound. Value Probl., 2020, Article 102 pp. (2020) · Zbl 1487.35189 |
| [20] | Li, Y. A.; Olver, P., Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differ. Equ., 162, 27-63 (2000) · Zbl 0958.35119 |
| [21] | Ma, F.; Liu, Y.; Qu, C., Wave-breaking phenomena for the nonlocal Whitham-type equations, J. Differ. Equ., 261, 6029-6054 (2016) · Zbl 1350.35038 |
| [22] | Whitham, G. B., Variational methods and applications to water waves, Proc. R. Soc. A, 299, 6-25 (1967) · Zbl 0163.21104 |
| [23] | Wei, L.; Wang, Y.; Zhang, H., Breaking waves and persistence property for a two-component Camassa-Holm system, J. Math. Anal. Appl., 445, 1084-1096 (2017) · Zbl 1352.35129 |
| [24] | Wei, L., Wave breaking analysis for the Fornberg-Whitham equation, J. Differ. Equ., 265, 2886-2896 (2018) · Zbl 1394.35377 |
| [25] | Wei, L., New wave-breaking criteria for the Fornberg-Whitham equation, J. Differ. Equ., 280, 571-589 (2021) · Zbl 1461.35192 |
| [26] | Wu, X. L.; Zhang, Z., On the blow-up of solutions for the Fornberg-Whitham equation, Nonlinear Anal., Real World Appl., 44, 573-588 (2018) · Zbl 1404.35066 |
| [27] | Yang, S. J., Wave breaking phenomena for the Fornberg-Whitham equation, J. Dyn. Differ. Equ., 33, 1753-1758 (2020) · Zbl 1478.35056 |
| [28] | Yin, Z., On the Cauchy problem for an integrable equation with peakon solutions, Ill. J. Math., 47, 3, 649-666 (2003) · Zbl 1061.35142 |
| [29] | Zhou, Y., Blow-up of solutions to a nonlinear dispersive rod equation, Calc. Var. Partial Differ. Equ., 25, 63-77 (2006) · Zbl 1172.35504 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
