A remark on global well-posedness below $L^2$ for the GKDV-3 equation

Abstract

The $I$-method in its first version as developed by Colliander et~al. in [2] is applied to prove that the Cauchy-problem for the generalized Korteweg-de Vries equation of order three (gKdV-3) is globally well-posed for large real-valued data in the Sobolev space $H^s(\mathbb R \rightarrow \mathbb R)$, provided $s>-\frac{1}{42}$.