Advanced Search
Article Contents
Article Contents

The existence of weak solutions for a generalized Camassa-Holm equation

Abstract Related Papers Cited by
  • A Camassa-Holm type equation containing nonlinear dissipative effect is investigated. A sufficient condition which guarantees the existence of weak solutions of the equation in lower order Sobolev space $H^s$ with $1 \leq s \leq \frac{3}{2}$ is established by using the techniques of the pseudoparabolic regularization and some prior estimates derived from the equation itself.
    Mathematics Subject Classification: Primary: 35Q53; Secondary: 35L05.


    \begin{equation} \\ \end{equation}
  • [1]

    R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.doi: doi:10.1103/PhysRevLett.71.1661.


    Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.doi: doi:10.1006/jdeq.1999.3683.


    Z. H. Guo, M. Jiang, Z. Wang and G. F. Zheng, Global weak solutions to the Camassa-Holm equation, Discrete and Continuous Dynamical Systems, 21 (2008), 883-906.doi: doi:10.3934/dcds.2008.21.883.


    A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.doi: doi:10.1007/s002200050801.


    R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444.doi: doi:10.1016/S0022-0396(03)00096-2.


    A. M. Wazwaz, A class of nonlinear fourth order variant of a generalized Camassa-Holm equation with compact and noncompact solutions, Appl. Math. Comput., 165 (2005), 485-501.doi: doi:10.1016/j.amc.2004.04.029.


    A. M. Wazwaz, The tanh-coth and the sine-cosine methods for kinks, solitons and periodic solutions for the Pochhammer-Chree equations, Appl. Math. Comput., 195 (2008), 24-33.doi: doi:10.1016/j.amc.2007.04.066.


    L. X. Tian and X. Y. Song, New peaked solitary wave solutions of the generalized Camassa-Holm equation, Chaos, Solitons and Fractals, 19 (2004), 621-637.doi: doi:10.1016/S0960-0779(03)00192-9.


    Y. Zheng and S. Y. Lai, Peakons, solitary patterns and periodic solutions for generalized Camassa-Holm equations, Phys. Lett. A, 372 (2008), 4141-4143.doi: doi:10.1016/j.physleta.2007.03.096.


    S. Hakkaev and K. Kirchev, Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation, Communications in Partial Differential Equations, 30 (2005), 761-781.doi: doi:10.1081/PDE-200059284.


    T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equation, Commun. Pure Appl. Math., 41 (1988), 891-907.doi: doi:10.1002/cpa.3160410704.


    J. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation, Phil. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601.doi: doi:10.1098/rsta.1975.0035.


    A. Moameni, Soliton solutions for quasilinear Schr$\ddoto$dinger equations involving supercritical exponent in $R^N$, Communications on Pure and Applied Analysis, 7 (2008), 89-105.doi: doi:10.3934/cpaa.2008.7.89.


    R. M. Colombo and G. Guerra, Hyperbolic balance laws with a dissipative non local source, Communications on Pure and Applied Analysis, 7 (2008), 1077-1090.doi: doi:10.3934/cpaa.2008.7.1077.


    D. Pilod, Sharp well-posedness results for the Kuramoto-Velarde equation, Communications on Pure and Applied Analysis, 7 (2008), 867-881.doi: doi:10.3934/cpaa.2008.7.867.


    K. Y. Wang, Global well-posedness for a transport equation with non-local velocity and critical diffusion, Communications on Pure and Applied Analysis, 7 (2008), 1203-1210.doi: doi:10.3934/cpaa.2008.7.1203.


    C. L. He, D. X. Kong and K. F. Liu, Hyperbolic mean curvature flow, J. Differential Equations, 246 (2009), 373-390.doi: doi:10.1016/j.jde.2008.06.026.

  • 加载中

Article Metrics

HTML views() PDF downloads(74) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint