Advances in Differential Equations

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

Seungly Oh

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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.

Article information

Source
Adv. Differential Equations Volume 18, Number 7/8 (2013), 633-662.

Dates
First available in Project Euclid: 20 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1369057709

Mathematical Reviews number (MathSciNet)
MR3086670

Zentralblatt MATH identifier
1291.35303

Subjects
Primary: 5Q53 35B10: Periodic solutions 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx]

Citation

Oh , Seungly. Resonant phase-shift and global smoothing of the periodic Korteweg-de Vries equation in low regularity settings. Adv. Differential Equations 18 (2013), no. 7/8, 633--662. https://projecteuclid.org/euclid.ade/1369057709.


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