Differential and Integral Equations

Regularity and uniqueness for the rough solutions of the derivative nonlinear Schrödinger equation

Yuanyuan Dan, Yongsheng Li, and Cui Ning

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Abstract

In this paper, we obtain the unconditional uniqueness for the rough solutions of the derivative nonlinear Schrödinger equation $$ i\partial_t u + \partial^2_{x} u =i\partial_{x}(|u|^2u) $$ in $C([0,T];H^s(\mathbb R))$, $s\in(\frac{2}{3},1]$. The arguments used here are the normal form argument, resonant decomposition and the Bourgain argument. The main ingredient in the proof is to improve the regularity of the solution by iteration method and finally show that the solution belongs to some Bourgain space.

Article information

Source
Differential Integral Equations, Volume 32, Number 5/6 (2019), 265-290.

Dates
First available in Project Euclid: 3 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.die/1554256867

Mathematical Reviews number (MathSciNet)
MR3938340

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 35A01: Existence problems: global existence, local existence, non-existence

Citation

Dan, Yuanyuan; Li, Yongsheng; Ning, Cui. Regularity and uniqueness for the rough solutions of the derivative nonlinear Schrödinger equation. Differential Integral Equations 32 (2019), no. 5/6, 265--290. https://projecteuclid.org/euclid.die/1554256867


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