Non-existence of solutions for the periodic cubic NLS below L^2

Tadahiro Oh; Zihua Guo

doi:10.1093/imrn/rnw271

Non-existence of solutions for the periodic cubic NLS below L^2

Tadahiro Oh, Zihua Guo

School of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We prove non-existence of solutions for the cubic nonlinear Schrödinger equation (NLS) on the circle if initial data belong to H^s(핋)\setminus L^2(핋) for some s∈(−1/8,0). The proof is based on establishing an a priori bound on solutions to a renormalized cubic NLS in negative Sobolev spaces via the short time Fourier restriction norm method.

Original language	English
Pages (from-to)	1656-1729
Number of pages	74
Journal	International Mathematics Research Notices
Volume	2018
Issue number	6
Early online date	26 Dec 2016
DOIs	https://doi.org/10.1093/imrn/rnw271
Publication status	Published - 20 Mar 2018

Keywords

nonlinear Schrödinger equation
well-posedness
ill-posedness
short time Fourier restriction norm method
a priori estimate
normal form reduction

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10.1093/imrn/rnw271

Non-existence of solutions for the periodic cubic NLS below L^2
This is a pre-copyedited, author-produced PDF of an article accepted for publication in International Mathematics Research Notices following peer review.
Accepted author manuscript, 605 KB

ProbDynDispEq - Probabilistic and Dynamical Study of Nonlinear Dispersive Equations
Oh, T.
EU government bodies
1/03/15 → 29/02/20
Project: Research

Cite this

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title = "Non-existence of solutions for the periodic cubic NLS below L^2",

abstract = "We prove non-existence of solutions for the cubic nonlinear Schr{\"o}dinger equation (NLS) on the circle if initial data belong to H^s(핋)\setminus L^2(핋) for some s∈(−1/8,0). The proof is based on establishing an a priori bound on solutions to a renormalized cubic NLS in negative Sobolev spaces via the short time Fourier restriction norm method. ",

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AU - Guo, Zihua

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N2 - We prove non-existence of solutions for the cubic nonlinear Schrödinger equation (NLS) on the circle if initial data belong to H^s(핋)\setminus L^2(핋) for some s∈(−1/8,0). The proof is based on establishing an a priori bound on solutions to a renormalized cubic NLS in negative Sobolev spaces via the short time Fourier restriction norm method.

AB - We prove non-existence of solutions for the cubic nonlinear Schrödinger equation (NLS) on the circle if initial data belong to H^s(핋)\setminus L^2(핋) for some s∈(−1/8,0). The proof is based on establishing an a priori bound on solutions to a renormalized cubic NLS in negative Sobolev spaces via the short time Fourier restriction norm method.

KW - nonlinear Schrödinger equation

KW - well-posedness

KW - ill-posedness

KW - short time Fourier restriction norm method

KW - a priori estimate

KW - normal form reduction

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JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

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Non-existence of solutions for the periodic cubic NLS below L^2

Abstract

Keywords

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Projects

ProbDynDispEq - Probabilistic and Dynamical Study of Nonlinear Dispersive Equations

Cite this