On the periodic Korteweg-de Vries equation: a normal form approach

This paper discusses an improved smoothing phenomena for low-regularity solutions of the Korteweg-de Vries (KdV) equation in the periodic settings by means of normal form transformation. As a result, the solution map from a ball on $H^{-1\mathrm{/}2+}$ to $C_0^t([0,T],H^{-1\mathrm{/}2+})$ can be shown to be Lipschitz in a $H_x^{0+}$ topology, where the Lipschitz constant only depends on the rough norm $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H^{-1\mathrm{/}2+}}$ of the initial data. A similar episode has been observed in a recent paper on 1D quadratic Schr"odinger equation in low-regularity setting.