In this paper, we consider an asymptotic model for wave propagation in shallow water with the effect of the Coriolis force is derived from the governing equation in two dimensional flows. Motivated by the eariler works (Brandolese in Int Math Res Not 22:5161–5181, 2012; Escauriaza et al. in J Funct Anal 244:504–535, 2007; Himonas et al. in Commun Math Phys 271:511–522, 2007; Himonas and Misiolek in Math Ann 327:575–584, 2003; Kohlmann in Z Angew Math Mech 94:264–272, 2014), we demonstrate the persistence results for the solution in weighted \(L^p\) spaces for a large classs of moderate weights. We also discuss the spatial asymptotic profiles of solutions to this model equation. Finally, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Aldroubi, A., Gröchenig, K.: Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev. 43, 585–620 (2001)
Baouendi, M.S., Goulaouic, C.: Remarks on the abstract form of nonlinear Cauchy–Kowalevski theorems. Commun. Partial Differ. Equ. 2, 1151–1162 (1977)
Baouendi, M.S., Goulaouic, C.: Sharp estimates for analytic pseudodifferential operators and applications to Cauchy problems. J. Differ. Equ. 48, 241–268 (1983)
Benjamin, T., Bona, J., Mahony, J.: Model equations for long waves in nonlinear dispersive media. Philos. Trans. R. Soc. Lond. A 272, 47–78 (1972)
Brandolese, L.: Breakdown for the Camassa–Holm equation using decay criteria and persistence in weighted spaces. Int. Math. Res. Not. 22, 5161–5181 (2012)
Busuioc, V.: On second grade fluids with vanishing viscosity. Comput. Rend. Acad. Sci. Ser. I-Math. 328, 1241–1246 (1999)
Bona, J.L., Tzvetkov, N.: Sharp well-posedness results for the BBM equation. Discrete Cont. Dyn. Syst. Ser. A 23, 1241–1252 (2009)
Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked soliton. Phys. Rev. Lett. 71, 1661–1664 (1993)
Camassa, R., Holm, D.D., Hyman, J.M.: A new integral shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994)
Chen, M., Gui, G., Liu, Y.: On a shallow-water approximation to the Green-Naghdi equations with the Coriolis effect. Adv. Math. 340, 106–137 (2018)
Constantin, A.: Nonlinear water waves with applications to wave-current interactions and Tsunamis. CBMS-NSF Regional Conference Series in Applied Mathematics, 81 SIAM, Philadelphia (2012)
Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)
Constantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26, 303–328 (1998)
Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192, 165–186 (2009)
Constantin, A., Johnson, R.S.: The dynamics of waves interacting with the Equatorial Undercurrent. Geophys. Astrophys. Fluid Dyn. 109, 311–358 (2015)
Constantin, A., Ivanov, R.I.: Equatorial wave-current interactions. Commun. Math. Phys. 370, 1–48 (2019)
Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)
Constantin, A., Escher, J.: Particle trajectories in solitary water waves. Bull. Am. Math. Soc. 44, 423–431 (2007)
Dai, H.H.: Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod. Acta Mech. 127, 193–207 (1998)
Degasperis, A., Holm, D.D., Hone, A.N.W.: A new integrable equation with peakon solutions. Theor. Math. Phys. 133, 1463–1474 (2002)
Degasperis, A., Procesi, M.: Asymptotic integrability. In: Degasperis, A., Gaeta, G. (eds.) Symmetry and Perturbation Theory, pp. 23–37. World Scientific, River Edge, New Jersey (1999)
Escauriaza, L., Kenig, C.E., Ponce, G., Vega, L.: On uniqueness properties of solutions of the k-generalized KdV equations. J. Funct. Anal. 244(2), 504–535 (2007)
Fan, L., Gao, H., Liu, Y.: On the rotation-two-component Camassa–Holm system modeling the equatorial water waves. Adv. Math. 291, 59–89 (2016)
Fuchssteiner, B., Fokas, A.S.: Symplectic structures, their bäcklund transformations and hereditary symmetries. Phys. D 4, 47–66 (1981)
Geyer, A., Quirchmayr, R.: Shallow water equations for equatorial tsunami waves. Philos. Trans. R. Soc. A 376, 20170100 (2018)
Himonas, A.A., Misiolek, G., Ponce, G., Zhou, Y.: Persistence properties and unique continuation of solutions of the Camassa-Holm equation. Commun. Math. Phys. 271(2), 511–522 (2007)
Himonas, A.A., Misiolek, G.: Analyticity of the Cauchy problem for an integrable evolution equation. Math. Ann. 327, 575–584 (2003)
Ionescu-Kruse, D.: Variational derivation of a geophysical Camassa-Holm type shallow water equation. Nonlinear Anal. 156, 286–294 (2017)
Ivanov, R.: Hamiltonian model for coupled surface and internal waves in the presence of currents. Nonlinear Anal. Real World Appl. 34, 316–334 (2017)
Johnson, R.S.: Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 455, 63–82 (2002)
Kohlmann, M.: The two-component Camassa-Holm system in weighted \(L_p\) spaces. Z. Angew. Math. Mech. 94(3), 264–272 (2014)
Kano, K., Nishida, T.: A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves. Osaka J. Math 23, 389–413 (1986)
Korteweg, D.J., de Vries, G.: On the change of the form of long waves advancing in rectangular channel, and on a new type of long stationary waves. Philos. Mag. 39, 422–443 (1895)
Lannes, D.: The Water Waves Problem. Mathematical Analysis and Asymptotics, Mathematical Surveys and Monographs, vol. 188. American Mathematical Society, Providence (2013)
Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467–490 (1968)
Lenells, J.: Traveling wave solutions of the Camassa–Holm equation. J. Differ. Equ. 217, 393–430 (2005)
Luo, T., Liu, Y., Mi, Y., Moon, B.: On a shallow-water model with the Coriolis effect. (2018, submitted)
Miura, R.M.: The Korteweg-de Vries equation: a survey of results. SIAM Rev. 18, 412–459 (1976)
Ni, L., Zhou, Y.: A new asymptotic behavior for solutions of the Camassa-Holm equation. Proc. Am. Math. Soc. 140, 607–614 (2012)
Toland, J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)
Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)
Yan, K., Yin, Z.: Analytic solutions of the Cauchy problem for two-component shallow water system. Math. Z. 269, 1113–1127 (2011). https://doi.org/10.1007/s00209-010-0775-5
Yan, K., Yin, Z.: Analyticity of the Cauchy problem for two-component Hunter-Saxton systems. Nonlinear Anal. 75, 253–259 (2012)
Zhou, S.: Persistence properties for a generalized Camassa-Holm equation in weighted \(L^p\) spaces. J. Math. Anal. Appl. 410, 932–938 (2014)
This work was supported by Incheon National University Research Grant 2017-0087 and Basic Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2017R1C1B1002336).
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Communicated by Adrian Constantin.
About this article
Cite this article
Moon, B. Persistence property and analyticity for a shallow-water model with the coriolis effect in weighted spaces. Monatsh Math (2021). https://doi.org/10.1007/s00605-021-01523-x
- Shallow water
- asymptotic model
- Coriolis force
- Green-Naghdi equations
- Weighted space
Mathematics Subject Classification