Local well-posedness of the Cauchy problem for the generalized Camassa–Holm equation in Besov spaces

Volume 34 / 2007

Gang Wu, Jia Yuan Applicationes Mathematicae 34 (2007), 253-267 MSC: Primary 35G25; Secondary 35Q35. DOI: 10.4064/am34-3-1

Abstract

We study local well-posedness of the Cauchy problem for the generalized Camassa–Holm equation $\partial_{t}u-\partial^{3}_{txx}u+2\kappa\partial_{x}u+\partial_{x}[{g(u)}/{2}] =\gamma(2\partial_{x}u\partial^{2}_{xx}u+u\partial^{3}_{xxx}u)$ for the initial data $u_{0}(x)$ in the Besov space $B^{s}_{p,r}(\Bbb R)$ with $\max({3}/{2},1 +{1}/{p})< s\leq m$ and $(p,r)\in [1,\infty]^{2}$, where $g:\Bbb R\rightarrow\Bbb R$ is a given $C^{m}$-function ($m\geq 4$) with $g(0)=g'(0)=0$, and $\kappa\geq 0$ and $\gamma\in \Bbb R$ are fixed constants. Using estimates for the transport equation in the framework of Besov spaces, compactness arguments and Littlewood–Paley theory, we get a local well-posedness result.

Authors

  • Gang WuThe Graduate School of China Academy of Engineering Physics
    P.O. Box 2101, Beijing 100088, China
    e-mail
  • Jia YuanThe Graduate School of China Academy of Engineering Physics
    P.O. Box 2101, Beijing 100088, China
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image