On well-posedness and wave operator for the gKdV equation

Abstract

We consider the generalized Korteweg-de Vries (gKdV) equation ${\partial }_{t}u+{\partial }_x^{3}u+\mu {\partial }_x(u^{k+1})=0$, where $k>4$ is an integer number and $\mu =\pm 1$. We give an alternative proof of the Kenig, Ponce, and Vega result in \cite{kpv1}, which asserts local and global well-posedness in ${\dot{H}}^{s_k}(\mathbb{R})$, with $s_k=(k-4)/2k$. A blow-up alternative in suitable Strichatz-type spaces is also established. The main tool is a new linear estimate. As a consequence, we also construct a wave operator in the critical space ${\dot{H}}^{s_k}(\mathbb{R})$, extending the results of Côte [2].

11 pages. To appear Bulletin des Sciences Mathématiques

References in corpus (1)

• Long range scattering for some Schr"odinger related nonlinear systems