Analyticity of the Cauchy Problem for a Three-Component Generalization of Camassa–Holm Equation
Abstract
:1. Introduction
2. Analytic Solutions to the System (1)
- (i)
- If for any the function is holomorphic in and continuous on with values in and then is a holomorphic functionon with values in
- (ii)
- For any and any that is, we have
- (iii)
- exists such that for any
3. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Geng, X.G.; Xue, B. A three-component generalization of Camassa–Holm equation with N-peakon solutions. Adv. Math. 2011, 226, 827–839. [Google Scholar] [CrossRef]
- Camassa, R.; Holm, D.D. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 1993, 71, 1661–1664. [Google Scholar] [CrossRef] [PubMed]
- Popowicz, Z. A Camassa–Holm equation interacted with the Degasperis-Procesi equation. Czech. J. Phys. 2006, 56, 1263–1268. [Google Scholar] [CrossRef]
- Fuchssteiner, B. Some tricks from the symmetry-tool box for nonlinear equations: Generalizations of the Camassa–Holm equation. Phys. D 1996, 95, 229–243. [Google Scholar] [CrossRef]
- Johnson, R.S.; Holm, C. Korteweg de Vries and related models for water waves. J. Fluid Mech. 2002, 455, 63–82. [Google Scholar] [CrossRef]
- Fokas, A.; Fuchssteiner, B. Symplectic structures, their Bäcklund transformation and hereditary symmetries. Phys. D 1981, 4, 47–66. [Google Scholar]
- Dai, H. Model equations for nolinear dispersive waves in compressible Mooney-Rivlin rod. Acta Mech. 1998, 127, 193–207. [Google Scholar] [CrossRef]
- Constantin, A.; Escher, J. Well-posedness; global existence, and blow up phenomena for a periodic quasilinear hyperbolic equation. Comm. Pure Appl. Math. 1998, 51, 475–504. [Google Scholar] [CrossRef]
- Constantin, A.; Escher, J. Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 1998, 181, 229–243. [Google Scholar] [CrossRef]
- Xin, Z.; Zhang, P. On the weak solutions to a shallow water equation. Comm. Pure Appl. Math. 2000, 53, 1411–1433. [Google Scholar] [CrossRef]
- Xin, Z.; Zhang, P. On the uniqueness and large time behavior of the weak solution to a shallow water equation. Commun. Partial Diff. Eqs. 2002, 27, 1815–1844. [Google Scholar] [CrossRef]
- Bressan, A.; Constantin, A. Global dissipative solutions of the Camassa–Holm equation. Anal. Appl. 2007, 5, 1–27. [Google Scholar] [CrossRef]
- Constantin, A.; Ivanov, R. On an integrable two-component Camassa–Holm shallow water system. Phys. Lett. A 2008, 372, 7129–7132. [Google Scholar] [CrossRef]
- Yan, K.; Yin, Z. Initial boundary value problems for the two-component shallow water systems. Rev. Mat. Iberoam. 2013, 29, 911–938. [Google Scholar] [CrossRef]
- Yan, K.; Yin, Z. On the solutions of the Dullin-Gottwald-Holm equation in Besov spaces. Nonlinear Anal. RWA 2012, 13, 2580–2592. [Google Scholar] [CrossRef]
- Zhang, Z.Y.; Liu, Z.; Deng, Y.; Huang, C.; Lin, S.; Zhu, W. Global well-posedness and infinite propagation speed for the N-abc family of Camassa–Holm type equation with both dissipation and dispersion. J. Math. Phys. 2020, 61, 071502. [Google Scholar] [CrossRef]
- Zhang, Z.Y.; Li, L.; Fang, C.; He, F.; Huang, C.; Zhu, W. A new blow-up criterion for the N-abc family of Camassa–Holm type equation with both dissipation and dispersion. Open Math. 2020, 18, 194–203. [Google Scholar] [CrossRef]
- Zhang, Z.Y.; Huang, J.H.; Sun, M.B. Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm equation revisited. J. Math. Phys. 2015, 56, 092703. [Google Scholar] [CrossRef]
- Zhang, Z.Y.; Liu, Z.H.; Deng, Y.J. Global energy conservation solution for the N-abc family of Camassa–Holm type equation. Nonlinear Anal. RWA 2024, 78, 104093. [Google Scholar] [CrossRef]
- Bressan, A.; Constantin, A. Global conservative solutions to the Camassa–Holm equation. Arch. Ration. Mech. Anal. 2007, 183, 215–239. [Google Scholar] [CrossRef]
- Chen, G.; Chen, M.; Liu, Y. Existence and uniqueness of the global conservative weak solutions for the integrable Novikov equation. Indiana Univ. Math. J. 2018, 67, 2393–2433. [Google Scholar] [CrossRef]
- Luo, W.; Yin, Z. Global existence and local well-posedness for a three-component Camassa–Holm system with N-peakon solutions. J. Differ. Equ. 2015, 259, 201–234. [Google Scholar] [CrossRef]
- Escher, J.; Lechtenfeld, O.; Yin, Z. Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation. Discrete Contin. Dyn. Syst. 2007, 19, 493–513. [Google Scholar] [CrossRef]
- Guan, C.; Yin, Z. Global existence and blow-up phenomena for an integrable two-component Camassa–Holm shallow water system. J. Differ. Equ. 2010, 248, 2003–2014. [Google Scholar] [CrossRef]
- Guan, C.; Yin, Z. Global weak solutions for a two-component Camassa–Holm shallow water system. J. Func. Anal. 2011, 260, 1132–1154. [Google Scholar] [CrossRef]
- Gui, G.; Liu, Y. On the global existence and wave-breaking criteria for the two-component Camassa–Holm system. J. Func. Anal. 2010, 258, 4251–4278. [Google Scholar] [CrossRef]
- Gui, G.; Liu, Y. On the Cauchy probelm for the two-component Camassa–Holm system. Math. Z. 2011, 268, 45–66. [Google Scholar] [CrossRef]
- Holm, D.; Naraigh, L.; Tronci, C. Singular solution of a modified two-component Camassa–Holm equation. Phys. Rev. E 2009, 79, 1–13. [Google Scholar] [CrossRef]
- Guan, C.; Yin, Z. Global weak solutions for a modified two-component Camassa–Holm equation. Ann. Inst. Henri Poincaré C 2011, 28, 623–641. [Google Scholar] [CrossRef]
- Tan, W.; Yin, Z. Global conservative solutions of a modified two-component Camassa–Holm shallow water system. J. Differ. Equ. 2011, 251, 3558–3582. [Google Scholar] [CrossRef]
- Tan, W.; Yin, Z. Global dissipative solutions of a modified two-component Camassa–Holm shallow water system. J. Math. Phys. 2011, 52, 033507. [Google Scholar] [CrossRef]
- Wang, Y.J.; Song, Y.D. On the global existence of dissipative solutions for the modified coupled Camassa–Holm system. Soft Comput. 2013, 17, 2007–2019. [Google Scholar] [CrossRef]
- Ovsiannikov, L.V. Non-local Cauchy problems in fluid dynamics. Actes Congress Int. Math. Nice 1970, 3, 137–142. [Google Scholar]
- Ovsiannikov, L.V. A nonlinear Cauchy problems in a scale of Banach spaces. Dokl. Akad. Nauk SSSR 1971, 12, 1497–1502. [Google Scholar]
- Constantin, A.; Escher, J. Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 2011, 173, 559–568. [Google Scholar] [CrossRef]
- Nirenberg, L. An abstract form of the nonlinear Cauchy–Kowalevski theorem. J. Differ. Geom. 1972, 6, 561–576. [Google Scholar]
- Nishida, T. A note on a theorem of Nirenberg. J. Differ. Geom. 1977, 12, 629–633. [Google Scholar] [CrossRef]
- Baouendi, S.; Goulaouic, C. Remarks on the abstract form of nonlinear Cauchy–Kowalevski theorems. Comm. PDE 1977, 2, 1151–1162. [Google Scholar] [CrossRef]
- Baouendi, S.; Goulaouic, C. Sharp estimates for analytic pseudodifferential operators and applications to Cauchy problems. J. Differ. Equ. 1983, 48, 241–268. [Google Scholar] [CrossRef]
- Trubowitz, E. The inverse problem for periodic potentials. Comm. Pure Appl. Math. 1977, 30, 321–337. [Google Scholar] [CrossRef]
- Kato, T.; Masuda, K. Nonlinear evolution equations and analyticity I. Ann. Inst. Henri Poincaré C 1986, 3, 455–467. [Google Scholar] [CrossRef]
- Byers, P.; Himonas, A. Non-analytic solutions of the KdV equation. Abstr. Appl. Anal. 2004, 6, 453–460. [Google Scholar] [CrossRef]
- Bona, J.; Smith, R. The initial value problem for the Korteweg-de Vries equation. Phil. Trans. Roy. Soc. Lond. A 1975, 278, 555–601. [Google Scholar] [CrossRef]
- Bourgain, J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II. KdV-Equation. Geom. Funct. Anal. 1993, 3, 107–156. [Google Scholar] [CrossRef]
- Kenig, C.; Ponce, G.; Vega, L. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math. 1993, 46, 527–620. [Google Scholar] [CrossRef]
- Kenig, C.; Ponce, G.; Vega, L. The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices. Duke Math. J. 1993, 71, 1–21. [Google Scholar] [CrossRef]
- McKean, H. Breakdown of a shallow water equation. Asian J. Math. 1998, 2, 867–874. [Google Scholar] [CrossRef]
- Himonas, A.A.; Misiolek, G. Analyticity of the Cauchy problem for an integrable evolution equation. Math. Ann. 2003, 327, 575–584. [Google Scholar] [CrossRef]
- Yan, K.; Yin, Z. Analytic solutions of the Cauchy problem for two-component shallow water systems. Math. Z. 2011, 269, 1113–1127. [Google Scholar] [CrossRef]
- Yan, K.; Yin, Z. On the initial value problem for higher dimensional Camassa–Holm equations. Discrete Contin. Dyn. Syst. 2015, 35, 1327–1358. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Shi, C.; Bin, M.; Zhang, Z. Analyticity of the Cauchy Problem for a Three-Component Generalization of Camassa–Holm Equation. Mathematics 2024, 12, 1085. https://doi.org/10.3390/math12071085
Shi C, Bin M, Zhang Z. Analyticity of the Cauchy Problem for a Three-Component Generalization of Camassa–Holm Equation. Mathematics. 2024; 12(7):1085. https://doi.org/10.3390/math12071085
Chicago/Turabian StyleShi, Cuiyun, Maojun Bin, and Zaiyun Zhang. 2024. "Analyticity of the Cauchy Problem for a Three-Component Generalization of Camassa–Holm Equation" Mathematics 12, no. 7: 1085. https://doi.org/10.3390/math12071085
APA StyleShi, C., Bin, M., & Zhang, Z. (2024). Analyticity of the Cauchy Problem for a Three-Component Generalization of Camassa–Holm Equation. Mathematics, 12(7), 1085. https://doi.org/10.3390/math12071085