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On the Cauchy problem for the generalized Camassa–Holm equation

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Abstract

In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation, which was proposed by Hakkaev and Kirchev (Comm Part Diff Equ 30:761–781, 2005). Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces \(H^{s},s>3/2\) for both the periodic and the nonperiodic case in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates. Finally, it is shown that the solution map for the generalized Camassa–Holm equation is Hölder continuous in \(H^{r}\)-topology.

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Acknowledgments

This work was partially supported by NSF of China (11371384), partially supported by NSF of Chongqing (cstc2013jcyjA0940) and partially supported by by NSF of Fuling (FLKJ,2013ABA2036).

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Authors and Affiliations

  1. College of Mathematics and Computer Sciences, Yangtze Normal University, Fuling, Chongqing, 408100, People’s Republic of China

    Yongsheng Mi

  2. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, People’s Republic of China

    Yongsheng Mi & Chunlai Mu

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  1. Yongsheng Mi
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  2. Chunlai Mu
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Correspondence to Yongsheng Mi.

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Communicated by A. Constantin.

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Mi, Y., Mu, C. On the Cauchy problem for the generalized Camassa–Holm equation. Monatsh Math 176, 423–457 (2015). https://doi.org/10.1007/s00605-014-0625-3

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  • DOI: https://doi.org/10.1007/s00605-014-0625-3

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