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

2012/06/18 by Seungly Oh, Oh, Seungly
Mathematics · #35B30 (Secondary) #35Q53 (Primary) 35B10 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B10 #msc:35B30 #msc:35Q53

paper · pdf · doi:10.48550/arxiv.1206.3898

arxiv created 2012/06/18 · arxiv updated 2012/06/19

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 [Su](t) - e^-t∂_x3 u(0) ∈ H^-s+1- where u(0) ∈ H^-s for 0≤ s <1/2 where S is the resonant phase-shift operator described below. We use the normal form method to obtain the result.

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