#### Vol. 9, No. 3, 2016

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On the well-posedness of the generalized Korteweg–de Vries equation in scale-critical $\hat{L}^r$-space

### Satoshi Masaki and Jun-ichi Segata

Vol. 9 (2016), No. 3, 699–725
##### Abstract

The purpose of this paper is to study local and global well-posedness of the initial value problem for the generalized Korteweg–de Vries (gKdV) equation in ${\stackrel{̂}{L}}^{r}=\left\{f\in {\mathsc{S}}^{\prime }\left(ℝ\right)\phantom{\rule{0.3em}{0ex}}:\phantom{\rule{0.3em}{0ex}}\parallel f{\parallel }_{{\stackrel{̂}{L}}^{r}}=\parallel \stackrel{̂}{f}{\parallel }_{{L}^{{r}^{\prime }}}<\infty \right\}$. We show (large-data) local well-posedness, small-data global well-posedness, and small-data scattering for the gKdV equation in the scale-critical ${\stackrel{̂}{L}}^{r}$-space. A key ingredient is a Stein–Tomas-type inequality for the Airy equation, which generalizes the usual Strichartz estimates for ${\stackrel{̂}{L}}^{r}$-framework.

##### Keywords
generalized Korteweg–de Vries equation, scattering problem
##### Mathematical Subject Classification 2010
Primary: 35Q53, 35B40
Secondary: 35B30