×

Perturbational blowup solutions to the 2-component Camassa-Holm equations. (English) Zbl 1236.35005

Summary: We study the perturbational method to construct the non-radially symmetric solutions of the compressible 2-component Camassa-Holm equations. In detail, we first combine the substitutional method and the separation method to construct a new class of analytical solutions for that system. In fact, we perturb the linear velocity \[ u = c(t)x+b(t), \] and substitute it into the system. Then, by comparing the coefficients of the polynomial, we can deduce the functional differential equations involving \((c(t),b(t),\rho^2(0,t))\). Additionally, we could apply Hubble’s transformation \(c(t) = \frac{\dot{a}(3t)}{a(3t)}\), to simplify the ordinary differential system involving \((a(3t),b(t),\rho^2(0,t))\). After proving the global or local existences of the corresponding dynamical system, a new class of analytical solutions is shown. To determine that the solutions exist globally or blow up, we just use the qualitative properties about the well-known Emden equation. Our solutions obtained by the perturbational method, fully cover M. Yuen’s solutions by the separation method [“Self-similar blowup solutions to the 2-component Camassa-Holm equations”, J. Math. Phys. 51, No. 9, Article No. 093524, 14p. (2010; doi:10.1063/1.3490189); erratum ibid. 52, No. 7, Article No. 079901, 1 p. (2011; doi:10.1063/1.3606591)].

MSC:

35B20 Perturbations in context of PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
35B44 Blow-up in context of PDEs
35B06 Symmetries, invariants, etc. in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Camassa, R.; Holm, D. D., Integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661-1664 (1993) · Zbl 0972.35521
[2] Chen, M.; Liu, S.-Q.; 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] Constantin, A., On the blow-up solutions of a periodic shallow water equation, J. Nonlinear Sci., 10, 391-399 (2000) · Zbl 0960.35083
[4] Constantin, A.; Ivanov, R., On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372, 7129-7132 (2008) · Zbl 1227.76016
[5] Deng, Y. B.; Xiang, J. L.; Yang, T., Blowup phenomena of solutions to Euler-Poisson equations, J. Math. Anal. Appl., 286, 295-306 (2003) · Zbl 1032.35023
[6] Escher, J.; Lechtenfeld, O.; Yin, Z., Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst. Ser. A, 19, 493-513 (2007) · Zbl 1149.35307
[7] Goldreich, P.; Weber, S., Homologously collapsing stellar cores, Astrophys. J., 238, 991-997 (1980)
[8] Guan, C. X.; Yin, Z. Y., Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system, J. Differential Equations, 248, 2003-2014 (2010) · Zbl 1190.35039
[9] Guo, Z. G., Blow-up and global solutions to a new integrable model with two components, J. Math. Anal. Appl., 372, 316-327 (2010) · Zbl 1205.35045
[10] Guo, Z. G.; Zhou, Y., On solutions to a two-component generalized Camassa-Holm system, Stud. Appl. Math., 124, 307-322 (2010) · Zbl 1189.35255
[11] Hale, J., Theory of Functional Differential Equations, Appl. Math. Sci., vol. 3 (1977), Springer-Verlag: Springer-Verlag New York-Heidelberg, x+365 pp
[12] Li, T. H., Some special solutions of the multidimensional Euler equations in \(R^N\), Commun. Pure Appl. Anal., 4, 757-762 (2005) · Zbl 1083.35058
[13] Li, T. H.; Wang, D. H., Blowup phenomena of solutions to the Euler equations for compressible fluid flow, J. Differential Equations, 221, 91-101 (2006) · Zbl 1083.76051
[14] Liang, Z. L., Blowup phenomena of the compressible Euler equations, J. Math. Anal. Appl., 379, 506-510 (2010) · Zbl 1196.35158
[15] Makino, T., Blowing up solutions of the Euler-Poisson equation for the evolution of the gaseous stars, Transport Theory Statist. Phys., 21, 615-624 (1992) · Zbl 0793.76069
[16] Makino, T., Exact solutions for the compressible Euler equation, J. Osaka Sangyo Univ. Natur. Sci., 95, 21-35 (1993)
[17] Walter, W., Ordinary Differential Equations, Grad. Texts Math. Read. Math., vol. 182 (1998), Springer-Verlag: Springer-Verlag New York, xii+380 pp., translated from the 6th German (1996) edition by Russell Thompson
[18] Whitham, G. B., Linear and Nonlinear Waves (1974), Wiley-Interscience: Wiley-Interscience New York-London-Sydney · Zbl 0373.76001
[19] Yuen, M. W., Blowup solutions for a class of fluid dynamical equations in \(R^N\), J. Math. Anal. Appl., 329, 1064-1079 (2007) · Zbl 1154.35427
[20] Yuen, M. W., Analytical blowup solutions to the 2-dimensional isothermal Euler-Poisson equations of gaseous stars, J. Math. Anal. Appl., 341, 445-456 (2008) · Zbl 1138.35364
[21] Yuen, M. W., Analytical solutions to the Navier-Stokes equations, J. Math. Phys., 49, 113102 (2008), 10 pp · Zbl 1159.81330
[22] Yuen, M. W., Analytical blowup solutions to the pressureless Navier-Stokes-Poisson equations with density-dependent viscosity in \(R^N\), Nonlinearity, 22, 2261-2268 (2009) · Zbl 1173.85307
[23] Yuen, M. W., Self-similar blowup solutions to the 2-component Camassa-Holm equations, J. Math. Phys., 51, 093524 (2010), 14 pp · Zbl 1309.76042
[24] Yuen, M. W., Self-similar blowup solutions to the 2-component Degasperis-Procesi shallow water system, Commun. Nonlinear Sci. Numer. Simul., 16, 2993-2998 (2011) · Zbl 1419.76542
[25] Yuen, M. W., Perturbational blowup solutions to the 1-dimensional compressible Euler equations, preprint · Zbl 1254.76060
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.