×

Nonuniform continuity of the solution map to the two component Camassa-Holm system. (English) Zbl 1298.35182

Summary: We prove that the solution map of the two-component Camassa-Holm system is not uniformly continuous as a map from a bounded subset of the Sobolev space \(H^s(\mathbb T)\times H^r(\mathbb T)\) to \(C([0,1],H^s(\mathbb T)\times H^r(\mathbb T))\) when \(s\geqslant 1\) and \(r\geqslant 0\). We also demonstrate the nonuniform continuous property in the continuous function space \(C^1(\mathbb T)\times C^1(\mathbb T)\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Camassa, R.; Holm, D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661-1664 (1993) · Zbl 0972.35521
[2] Chen, M.; Liu, S.; Zhang, Y., A 2-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys., 75, 1-15 (2006) · Zbl 1105.35102
[3] Christ, M.; Colliander, J.; Tao, T., Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125, 1235-1293 (2003) · Zbl 1048.35101
[4] Constantin, A.; Escher, J., Well-posedness, global existence and blow-up phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51, 475-504 (1998) · Zbl 0934.35153
[5] Constantin, A.; Ivanov, R. I., On an integrable two-component Camassa Holm shallow water system, Phys. Lett. A, 372, 7129-7132 (2008) · Zbl 1227.76016
[6] Constantin, A.; Lannes, D., The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192, 165-186 (2009) · Zbl 1169.76010
[7] Constantin, A.; McKean, H. P., A shallow water equation on the circle, Comm. Pure Appl. Math., 52, 949-982 (1999) · Zbl 0940.35177
[8] Escher, J.; Lechtenfeld, O.; Yin, Z. Y., Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst., 19, 493-513 (2007) · Zbl 1149.35307
[9] Falqui, G., On a Camassa-Holm type equation with two dependent variables, J. Phys. A, 39, 327-342 (2006) · Zbl 1084.37053
[10] Fokas, A.; Fuchssteiner, B., Symplectic structures, their Bäcklund transformation and hereditary symmetries, Phys. D, 4, 47-66 (1981) · Zbl 1194.37114
[11] Gui, G.; Liu, Y., On the Cauchy problem for the two-component Camassa-Holm system, Math. Z. (2010)
[12] Himonas, A. A.; Kenig, C., Non-uniform dependence on initial data for the CH equation on the line, Differential Integral Equations, 22, 201-224 (2009) · Zbl 1240.35242
[13] Himonas, A. A.; Misiołek, G., High-frequency smooth solutions and well-posedness of the Camassa-Holm equation, Int. Math. Res. Not. IMRN, 51, 3135-3151 (2005) · Zbl 1092.35085
[14] Himonas, A. A.; Misiołek, G.; Ponce, G., Non-uniform continuity in \(H^1\) of the solution map of the CH equation, Asian J. Math., 11, 141-150 (2007) · Zbl 1127.35056
[15] Holm, D. D.; Nraigh, L.; Tronci, C., Singular solutions of a modified two-component Camassa-Holm equation, Phys. Rev. E, 79, 016601 (2009)
[16] Ivanov, R., Two-component integrable systems modelling shallow water waves: The constant vorticity case, Wave Motion, 46, 389-396 (2009) · Zbl 1231.76040
[17] Kato, T., The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58, 181-205 (1975) · Zbl 0343.35056
[18] 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., 46, 527-620 (1993) · Zbl 0808.35128
[19] Kenig, C.; Ponce, G.; Vega, L., On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106, 617-633 (2001) · Zbl 1034.35145
[20] Koch, H.; Tzvetkov, N., Nonlinear wave interactions for the Benjamin-Ono equation, Int. Math. Res. Not. IMRN, 30, 1833-1847 (2005) · Zbl 1156.35460
[21] Lenells, J., Traveling wave solutions of the Camassa-Holm equation, J. Differential Equations, 217, 393-430 (2005) · Zbl 1082.35127
[22] Misiołek, G., Classical solutions of the periodic Camassa-Holm equation, Geom. Funct. Anal., 12, 1080-1104 (2002) · Zbl 1158.37311
[23] Taylor, M. E., Partial Differential Equations I (1996), Springer-Verlag
[24] Thompson, R. C., The periodic Cauchy problem for the 2-component Camassa-Holm system, Differential Integral Equations, 26, 155-182 (2013) · Zbl 1299.35271
[25] Xin, Z. P.; Zhang, P., On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53, 1411-1433 (2000) · Zbl 1048.35092
[26] Zhang, P. Z.; Liu, Y., Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system, Int. Math. Res. Not. IMRN, 11, 1981-2021 (2010) · Zbl 1231.35184
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.