Abstract
Considered herein is the Cauchy problem of a model for shallow water waves of large amplitude. Using Littlewood–Paley decomposition and transport equation theory, we establish the local well-posedness of the equation in Besov spaces \(B^s_{p,r} \) with \(1\le p,r \le +\infty \) and \(s>\max \{1+\frac{1}{p },\frac{3}{2}\}\) (and also in Sobolev spaces \(H^s=B^s_{2,2}\) with \(s>3/2\)). Then, the precise blow-up mechanism for the strong solutions is determined in the lowest Sobolev space \(H^s \) with \(s>3/2\). Our results improve the corresponding work for this model in Quirchmayr (J Evol Equ 16:539–567, 2016), in which the Sobolev index \(s=3\) is required. In addition, we also investigate the asymptotic behaviors of the strong solutions to this equation at infinity within its lifespan provided the initial data lie in weighted \(L_{p,\phi }:=L_p(\mathbb {R},\phi ^pdx)\) spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Aldroubi and K. Gröchenig, Nonuniform sampling and reconstruction in shift-invariant spaces, SIAM Review, 43 (2001), 585–620.
H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, 2011.
T.B. Benjamin, J.L. Bona and J.J. Mahoney, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. A, 227 (1972), 47–78.
L. Brandolese, Breakdown for the Camassa–Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Not., 22 (2012), 5161–5181.
R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters 71 (1993), 1661–1664.
R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech. 31 (1994), 1–33.
A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321–362.
A. Constantin and J. Escher, Well-posedness, global existence and blow-up phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math. 51 (1998), 475–504.
A. Constantin, D. Lannes, The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165–186.
R. Danchin, Fourier analysis methods for PDEs, Lecture Notes, 14 November, 2003.
R. Danchin, A note on well-posedness for Camassa–Holm equation, J. Differential Equations, 192 (2003) 429–444.
N. Duruk Mutlubaş and A. Geyer, Orbital stability of solitary waves of moderate amplitude in shallow water, J. Differential Equations, 255 (2013), 254–263.
A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Phys. D, 4 (1981), 47–66.
A. Geyer, Solitary traveling water waves of moderate amplitude, J. Nonlinear Math. Phys., 19 (2012), 1240010.
R. S. Johnson, Camassa–Holm, Korteweg–de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63–82.
T. Kato, Quasi-linear equations of evolution with applications to partial differential equations, in: Spectral Theory and Differential Equations, in: Lecture Notes in Math., vol. 448, Springer-Verlag, Berlin, 1975, 25–70
D. J. Korteweg and G. de Vries On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422–443.
R. Quirchmayr, A new highly nonlinear shallow water wave equation, J. Evol. Equ., 16 (2016), 539–567.
S. Zhou, Persistence properties for a generalized Camassa–Holm equation in weighted \(L^p\) spaces, J. Math. Anal. Appl. 410 (2014), 932–938.
S. Zhou and C. Mu, Global conservative solutions for a model equation for shallow water waves of moderate amplitude, J. Differential Equations, 256 (2014), 1793–1816.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by University Young Core Teacher Foundation of Chongqing and the Talent Project of Chongqing Normal University (Grant No. 14CSBJ05), and Technology Research Foundation of Chongqing Educational Committee (Grant No. KJ1703043).
Rights and permissions
About this article
Cite this article
Zhou, S. Well-posedness and wave breaking for a shallow water wave model with large amplitude. J. Evol. Equ. 20, 141–163 (2020). https://doi.org/10.1007/s00028-019-00518-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-019-00518-4