Local well-posedness of the Cauchy problem for the generalized Camassa–Holm equation in Besov spaces
Volume 34 / 2007
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.
