×

On the Cauchy problem for the new integrable two-component Novikov equation. (English) Zbl 1441.35105

A new integrable two-component Novikov equation was derived by H. Li [J. Nonlinear Math. Phys. 26, No. 3, 390–403 (2019; Zbl 1417.37231)]. This system has the Lax pair formulation and can be expressed as a bi-Hamiltonian system.
The authors prove the local well-posedness of the two-component Novikov system in nonhomogeneous Besov spaces. They use the Littlewood-Paley theory and analysis of the transport equations.
Then, the authors verify that the blow-up may only occur in the form of the wave breaking implying that the solutions remain bounded but their first derivative may become unbounded in a finite time.
For analytic initial data, the unique solutions to the Cauchy problem remain analytic in both variables, globally in space and locally in time.
Finally, the authors prove that if the initial data decay exponentially or algebraically at infinity, the strong solutions of the two-component Novikov system also decay exponentially or algebraically at infinity within its lifespan.

MSC:

35G25 Initial value problems for nonlinear higher-order PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
35B44 Blow-up in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs

Citations:

Zbl 1417.37231
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baouendi, M.; Goulaouic, C., Sharp estimates for analytic pseudodifferential operators and application to the Cauchy problems, J. Differ. Equ., 48, 241-268 (1983) · Zbl 0535.35082
[2] Bona, J.; Smith, R., The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. R. Soc. Lond. Ser. A, 278, 555-601 (1975) · Zbl 0306.35027
[3] Bressan, A.; Constantin, A., Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5, 1-27 (2007) · Zbl 1139.35378
[4] Bressan, A.; Constantin, A., Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183, 215-239 (2007) · Zbl 1105.76013
[5] Chemin, J.: Localization in Fourier space and Navier-Stokes system, Phase Space Analysis of Partial Differential Equations. In: Proceedings, CRM series, Pisa, 53-136 (2004) · Zbl 1081.35074
[6] Constantin, A., Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 46, 023506 (2005) · Zbl 1076.35109
[7] Constantin, A., On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. A, 457, 953-970 (2001) · Zbl 0999.35065
[8] Constantin, A., The trajectories of particles in Stokes waves, Invent. Math., 166, 23-535 (2006) · Zbl 1108.76013
[9] Constantin, A., On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155, 352-363 (1998) · Zbl 0907.35009
[10] Constantin, A.; Escher, J., Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181, 229-243 (1998) · Zbl 0923.76025
[11] Constantin, A.; Escher, J., Particle trajectories in solitary water waves, Bull. Am. Math. Soc., 44, 423-431 (2007) · Zbl 1126.76012
[12] Constantin, A.; Escher, J., Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173, 559-568 (2011) · Zbl 1228.35076
[13] Constantin, A.; Gerdjikov, V.; Ivanov, RI, Inverse scattering transform for the Camassa-Holm equation, Inverse Probl., 22, 2197-2207 (2006) · Zbl 1105.37044
[14] Constantin, A.; Kappeler, T.; Kolev, B.; Topalov, T., On Geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom., 31, 155-180 (2007) · Zbl 1121.35111
[15] Constantin, A.; Lannes, D., The hydro-dynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 193, 165-186 (2009) · Zbl 1169.76010
[16] Constantin, A.; McKean, HP, A shallow water equation on the circle, Commun. Pure Appl. Math., 52, 949-982 (1999) · Zbl 0940.35177
[17] Camassa, R.; Holm, D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661-1664 (1993) · Zbl 0972.35521
[18] Constantin, A.; Ivanov, R.; Lenells, J., Inverse scattering transform for the Degasperis-Procesi equation, Nonlinearity, 23, 2559-2575 (2010) · Zbl 1211.37081
[19] Constantin, A.; Strauss, W., Stability of peakons, Commun. Pure Appl. Math., 53, 603-610 (2000) · Zbl 1049.35149
[20] Danchin, R., A few remarks on the Camassa-Holm equation, Differ. Integral Equ., 14, 953-988 (2001) · Zbl 1161.35329
[21] Danchin, R.: Fourier analysis methods for PDEs, Lecture Notes, 14 (2003) · Zbl 1048.35076
[22] Degasperis, A.; Holm, DD; Hone, ANW, A new integrable equation with peakon solutions, Theor. Math. Phys., 133, 1461-72 (2002)
[23] Degasperis, A.; Procesi, M.; Degasperis, A.; Gaeta, G., Asymptotic integrability, Symmetry and Perturbation Theory, 23-37 (1999), Singapore: World Scientific, Singapore · Zbl 0963.35167
[24] Fokas, A.; Fuchssteiner, B., Symplectic structures, their Backlund transformation and hereditary symmetries, Physica D, 4, 47-66 (1981) · Zbl 1194.37114
[25] Geng, X.; Xue, B., An extension of integrable peakon equations with cubic nonlinearity, Nonlinearity, 22, 1847-1856 (2009) · Zbl 1171.37331
[26] Gui, G.; Liu, Y., On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258, 4251-4278 (2010) · Zbl 1189.35254
[27] Grillakis, M.; Shatah, J.; Strauss, W., Stability theory of solitary waves in the presence of symmetry, J. Funct. Anal., 74, 160-197 (1987) · Zbl 0656.35122
[28] Himonas, A.; Mantzavinos, D., The initial value problem for a Novikov system, J. Math. Phys., 57, 071503 (2016) · Zbl 1342.35068
[29] Henry, D., Compactly supported solutions of the Camassa-Holm equation, J. Nonlinear Math. Phys., 12, 342-347 (2005) · Zbl 1086.35079
[30] Henry, D., Persistence properties for a family of nonlinear partial differential equations, Nonlinear Anal., 70, 1565-1573 (2009) · Zbl 1170.35509
[31] Himonas, A.; Misiolek, G., Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann., 327, 575-584 (2003) · Zbl 1073.35008
[32] Himonas, A.; Holliman, C., The Cauchy problem for the Novikov equation, Nonlinearity, 25, 449-479 (2012) · Zbl 1232.35145
[33] Himonas, A.; Holliman, C., The Cauchy problem for a generalized Camassa-Holm equation, Adv. Differ. Equ., 19, 161-200 (2014) · Zbl 1285.35093
[34] Himonas, A.; Misiołek, G., Non-uniform dependence on initial data of solutions to the Euler equations of hydrodynamics, Commun. Math. Phys., 296, 285-301 (2009) · Zbl 1195.35247
[35] Himonas, A.; Misiołlek, G.; Ponce, G.; Zhou, Y., Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Commun. Math. Phys., 271, 511-522 (2007) · Zbl 1142.35078
[36] Holden, H.; Raynaud, X., Dissipative solutions for the Camassa-Holm equation, Discret. Contin. Dyn. Syst., 24, 1047-1112 (2009) · Zbl 1178.65099
[37] Holden, H.; Raynaud, X., Global conservative solutions of the Camassa-Holm equations-a Lagrangianpoiny of view, Commun. Partial Differ. Equ., 32, 1511-1549 (2007) · Zbl 1136.35080
[38] Holm, DD; Staley, MF, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2, 323-380 (2003) · Zbl 1088.76531
[39] Home, ANW; Wang, JP, Integrable peakon equations with cubic nonlinearity, J. Phys. A, 41, 372002 (2008) · Zbl 1153.35075
[40] Hone, W.; Lundmark, H.; Szmigielski, J., Explicit multipeakon solutions of Novikov cubically nonlinear integrable Camassa-Holm type equation, Dyn. Partial Differ. Equ., 6, 253-289 (2009) · Zbl 1179.37092
[41] Hu, Q., Qiao, Z: Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function. arXiv:1511.03315 (2015)
[42] Jiang, Z.; Ni, L., Blow-up phenomenon for the integrable Novikov equation, J. Math. Anal. Appl., 385, 551-558 (2012) · Zbl 1228.35066
[43] Kato, T.; Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure Appl. Math., 41, 891-907 (1988) · Zbl 0671.35066
[44] Lai, S.; Wu, Y., The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation, J. Differ. Equ., 248, 2038-2063 (2010) · Zbl 1187.35179
[45] Lenells, J., Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl., 306, 72-82 (2005) · Zbl 1068.35163
[46] Li, H., Two-component generalizations of the Novikov equation, J. Nonlinear Math. Phys., 26, 390-403 (2019) · Zbl 1417.37231
[47] Liu, X.; Yin, Z., Local well-posedness and stability of peakons for a generalized Dullin-Gottwald-Holm equation, Nonlinear Anal., 74, 2497-2507 (2011) · Zbl 1211.35072
[48] Li, H.; Liu, QP, On bi-Hamiltonian structure of two-component Novikov equation, Phys. Lett. A, 377, 257-281 (2013) · Zbl 1298.37054
[49] Lundmark, H.; Szmigielski, J., Multi-peakon solutions of the Degasperis-Procesi equation, Inverse Probl., 21, 1553-1570 (2005) · Zbl 1086.35095
[50] Matsuno, Y., Multisoliton solutions of the Degasperis-Procesi equation and their peakon limit, Inverse Probl., 19, 1241-1245 (2003) · Zbl 1041.35090
[51] Misiolek, GA, Shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24, 203-208 (1998) · Zbl 0901.58022
[52] Ni, L.; Zhou, Y., Well-posedness and persistence properties for the Novikov equation, J. Differ. Equ., 250, 3002-3021 (2011) · Zbl 1215.37051
[53] Ni, L.; Zhou, Y., A new asymptotic behavior of solutions to the Camassa-Holm equation, Proc. Am. Math. Soc., 140, 607-614 (2012) · Zbl 1259.37046
[54] Novikov, V., Generalization of the Camassa-Holm equation, J. Phys. A, 42, 342002 (2009) · Zbl 1181.37100
[55] Taylor, M., Partial Differential Equations III, Nonlinear Equations (1996), Berlin: Springer, Berlin · Zbl 0869.35004
[56] Tiglay, F., The periodic Cauchy problem for Novikov equation, Int. Math. Res. Not., 2011, 4633-4648 (2011) · Zbl 1235.35011
[57] Toland, JF, Stokes waves, Topol Methods Nonlinear Anal., 7, 1-48 (1996) · Zbl 0897.35067
[58] Vakhnenko, VO; Parkes, EJ, Periodic and solitary-wave solutions of the Degasperis-Procesi equation, Chaos Solitons Fractals, 20, 1059-1073 (2004) · Zbl 1049.35162
[59] Wu, X.; Yin, Z., Well-posedness and global existence for the Novikov equation, Annali Sc. Norm. Sup. Pisa. X, I, 707-727 (2012) · Zbl 1261.35041
[60] Wu, X.; Yin, Z., Global weak solutions for the Novikov equation, J. Phys. A, 44, 055202 (2011) · Zbl 1210.35217
[61] Xia, B.; Qiao, Z., A new two-component integrable system with peakon solutions, Proc. R. Soc. A Math. Phys. Eng., 471, 20140750 (2015) · Zbl 1371.35265
[62] Xia, B.; Qiao, Z.; Zhou, R., A synthetical two-component model with Peakon solutions, Stud. Appl. Math., 135, 248-276 (2015) · Zbl 1338.35385
[63] Yan, K.; Qiao, Z.; Yin, Z., Qualitative analysis for a new integrable two-component Camassa-Holm system with peakon and weak kink solutions, Commun. Math. Phys., 336, 581-617 (2015) · Zbl 1317.35219
[64] Yan, W.; Li, Y.; Zhang, Y., Global existence and blow-up phenomena for the weakly dissipative Novikov equation, Nonlinear Anal., 75, 2464-2473 (2012) · Zbl 1258.35039
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.