NDLI: On the Cauchy problem for the generalized Camassa

Content Provider	Springer Nature Link
Author	Mu, Chunlai Mi, Yongsheng
Copyright Year	2014
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.
Ending Page	457
Page Count	35
Starting Page	423
File Format	PDF
ISSN	00269255
e-ISSN	14365081
Journal	Monatshefte für Mathematik
Issue Number	3
Volume Number	176
Language	English
Publisher	Springer Vienna
Publisher Date	2014-04-22
Publisher Place	Vienna
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Dependence of solutions on initial and boundary data, parameters KdV-like equations (Korteweg-de Vries) Cauchy-Kovalevskaya theorems Local well-posedness Mathematics Besov spaces Camassa–Holm type equation Initial value problems for nonlinear higher-order equations
Content Type	Text
Resource Type	Article
Subject	Mathematics

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On the Cauchy problem for the generalized Camassa–Holm equation