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

Gang Wu ¹   ; Jia Yuan ¹

¹ The Graduate School of China Academy of Engineering Physics P.O. Box 2101, Beijing 100088, China

Applicationes Mathematicae, Tome 34 (2007) no. 3, pp. 253-267

Résumé

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.

Détail

DOI : 10.4064/am34-3-1

Keywords: study local well posedness cauchy problem generalized camassa holm equation partial u partial txx kappa partial partial gamma partial partial partial xxx initial besov space bbb max leq infty where bbb rightarrow bbb given function geq kappa geq gamma bbb fixed constants using estimates transport equation framework besov spaces compactness arguments littlewood paley theory get local well posedness result