Resonant phase-shift and global smoothing of the periodic Korteweg-de Vries equation in low regularity settings

Abstract

We show a smoothing effect of near full-derivative for low-regularity global-in-time solutions of the periodic Korteweg--de Vries (KdV) equation. The smoothing is given by slightly shifting the space-time Fourier support of the nonlinear solution, which we call resonant phase-shift. More precisely, we show that $\mathcal{S}[u](t) - e^{-t{\partial}_x^3} u(0) \in H^{-s+1-}$, where $u(0) \in H^{-s}$ for $0\leq s < 1/2$ where $\mathcal{S}$ is the resonant phase-shift operator described below. We use the normal form method to obtain the result.