Wave breaking for the generalized Fornberg-Whitham equation. (English) Zbl 1543.76017
Summary: This paper aims to show that the Cauchy problem of the Burgers equation with a weakly dispersive perturbation involving the Bessel potential (generalization of the Fornberg-Whitham equation) can exhibit wave breaking for initial data with large slope. We also comment on the dispersive properties of the equation.
MSC:
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 76B25 | Solitary waves for incompressible inviscid fluids |
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q51 | Soliton equations |
Keywords:
Bessel potential; Cauchy problem; Burgers equation; weakly dispersive perturbation; solitary wave existence; global weak entropy solutionReferences:
| [1] | Adams, D. and Hedberg, L. I., Function Spaces and Potential Theory, , Springer-Verlag, Berlin, 1996. |
| [2] | Alonso-Orán, D., Durán, A., and Granero-Belinchón, R., Derivation and well-posedness for asymptotic models of cold plasmas, Nonlinear Anal., 244 (2024), 113539. · Zbl 1540.35472 |
| [3] | Albert, J., On the decay of solutions of the generalized Benjamin-Bona-Mahony equation, J. Math. Anal. Appl., 141 (1989), pp. 527-537. · Zbl 0697.35116 |
| [4] | Albert, J., Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation, J. Differential Equations, 63 (1986), pp. 117-134. · Zbl 0596.35109 |
| [5] | Camassa, R. and Holm, D., An integrable shallow water equation with peaked solitons, Phy. Rev. Lett., 71 (1973), pp. 1661-1664. · Zbl 0972.35521 |
| [6] | Castro, A., Córdoba, D., and Gancedo, F., Singularity formations for a surface wave model, Nonlinearity, 23 (2010), 2835. · Zbl 1223.35016 |
| [7] | Córdoba, D., Gómez-Serrano, J., and Ionescu, A., Global solutions for the generalized SQG patch equation, Arch. Ration. Mech. Anal., 233 (2019), pp. 1211-1251. · Zbl 1420.35424 |
| [8] | Constantin, A. and Escher, J., Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), pp. 229-243. · Zbl 0923.76025 |
| [9] | Deng, X., A note on wave-breaking criteria for the Fornberg-Whitham equation, Monatsh. Math., 202 (2023), pp. 93-102. · Zbl 1518.35142 |
| [10] | Ehrnström, M., Groves, M., and Wahlén, E., On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type, Nonlinearity, 25 (2012), pp. 2903-2936. · Zbl 1252.76014 |
| [11] | Fokas, A. and Fuchssteiner, B., Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981), pp. 47-66. · Zbl 1194.37114 |
| [12] | Fellner, K. and Schmeiser, C., Burgers-Poisson: A nonlinear dispersive model equation, SIAM J. Appl. Math., 64 (2004), pp. 1509-1525. · Zbl 1053.35092 |
| [13] | Fetecau, R. and Levy, D., Approximate model equations for water waves, Commun. Math. Sci., 3 (2005), pp. 159-170. · Zbl 1101.35065 |
| [14] | Fornberg, G. and Whitham, G., A numerical and theoretical study of certain nonlinear wave phenomena, Philos. Trans. Roy. Soc. London Ser. A, 289 (1978), pp. 373-404. · Zbl 0384.65049 |
| [15] | Germain, P., Pusateri, F., and Rousset, F., Asymptotic stability of solitons for mKdV, Adv. Math., 299 (2016), pp. 272-330. · Zbl 1348.35219 |
| [16] | Grunert, K. and Nguyen, K., On the Burgers-Poisson equation, J. Differential Equations, 261 (2016), pp. 3220-3246. · Zbl 1347.35139 |
| [17] | Haziot, S., Wave breaking for the Fornberg-Whitham equation, J. Differential Equations, 263 (2017), pp. 8178-8185. · Zbl 1375.35375 |
| [18] | Hörmann, G., Solution concepts, well-posedness and wave breaking for the Fornberg-Whitham equation, Monatsh. Math., 195 (2021), pp. 421-449. · Zbl 1467.35002 |
| [19] | Hur, V. M., Wave breaking for the Whitham equation, Adv. Math., 317 (2017), pp. 410-437. · Zbl 1375.35446 |
| [20] | Ivanov, R., On the integrability of a class of nonlinear dispersive equations, J. Nonlinear Math. Phys., 4 (2005), pp. 462-468. · Zbl 1089.35522 |
| [21] | Kwak, C. and Munoz, C., Extended decay properties for generalized BBM equations, in Nonlinear Dispersive Partial Differential Equations and Inverse Scattering, , Miller, P., Perry, P., Saut, J.-C., and Sulem, C., eds., Springer-Verlag, Berlin, 2019, pp. 397-412. |
| [22] | Lannes, D., Water Waves: Mathematical Theory and Asymptotics, , AMS, Providence, RI, 2013. · Zbl 1410.35003 |
| [23] | Linares, F., Pilod, D., and Saut, J.-C., Dispersive perturbations of Burgers and hyperbolic equations I: Local theory, SIAM J. Math. Anal., 46 (2014), pp. 1505-1537. · Zbl 1294.35124 |
| [24] | Ma, F., Liu, Y., and Qu, C., Wave-breaking phenomena for the nonlocal Whitham-type equations, J. Differential Equations, 261 (2016), pp. 6029-6054. · Zbl 1350.35038 |
| [25] | Mæhlen, O. and Xue, J., One-sided Hölder regularity of global weak solutions of negative order dispersive equations, J. Differential Equations, 364 (2023), pp. 412-455. · Zbl 1514.35077 |
| [26] | Naumkin, P. I. and Shishmarev, I. A., Nonlinear Nonlocal Equations in the Theory of Waves, , AMS, Providence, RI, 1994. · Zbl 0802.35002 |
| [27] | Nikol’skiĭ, S., Approximation of Functions of Several Variables and Imbedding Theorems, Springer-Verlag, Berlin, 1975. · Zbl 0307.46024 |
| [28] | Oh, S. and Pasqualotto, F., Gradient blow-up for dispersive and dissipative perturbations of the Burgers equation, Arch. Ration. Mech. Anal., 248 (2024), 54. · Zbl 1542.35345 |
| [29] | Saut, J.-C., Sur quelques généralisations de l’équation de Korteweg-deVries, J. Math. Pures Appl. (9), 58 (1979), pp. 21-61. · Zbl 0449.35083 |
| [30] | Saut, J.-C. and Wang, Y., Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity, Discrete Contin. Dyn. Syst., 41 (2021), pp. 1133-1155. · Zbl 1458.76016 |
| [31] | Saut, J.-C. and Wang, Y., Global dynamics of small solutions to the modified fractional Korteweg-de Vries and fractional cubic nonlinear Schrödinger equations, Comm. Partial Differential Equations, 46 (2021), pp. 1851-1891. · Zbl 1490.76046 |
| [32] | Saut, J.-C. and Wang, Y., The wave breaking for Whitham-type equations revisited, SIAM J. Math. Anal., 54 (2022), pp. 2295-2319. · Zbl 1486.76015 |
| [33] | Seliger, R., A note on the breaking of waves, Proc. Roy. Soc. Lond. Ser. A, 303 (1968), pp. 493-496. · Zbl 0159.28502 |
| [34] | Wang, Y., Global dynamics of the generalized fifth-order KdV equation with quintic nonlinearity, J. Evol. Equ., 21 (2021), pp. 1449-1475. · Zbl 1470.76023 |
| [35] | Whitham, G., Variational methods and applications to water waves, Proc. Roy. Soc. Lond. Ser. A, 299 (1967), pp. 6-25. · Zbl 0163.21104 |
| [36] | Whitham, G., Linear and Nonlinear Waves, John Wiley, New York, 1974. · Zbl 0373.76001 |
| [37] | Yang, S., Wave breaking phenomena for the Fornberg-Whitham equation, J. Dynam. Differential Equations, 33 (2021), pp. 1753-1758. · Zbl 1478.35056 |
| [38] | Yang, R., Shock formation of the Burgers-Hilbert equation, SIAM J. Math. Anal., 53 (2021), pp. 5756-5802. · Zbl 1475.35207 |
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.
