Norm Inflation for Nonlinear Schrödinger Equations in Fourier–Lebesgue and Modulation Spaces of Negative Regularity

We consider nonlinear Schrödinger equations in Fourier–Lebesgue and modulation spaces involving negative regularity. The equations are posed on the whole space, and involve a smooth power nonlinearity. We prove two types of norm inflation results. We first establish norm inflation results below the...

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Bibliographic Details
Published in:The Journal of fourier analysis and applications Vol. 26; no. 6
Main Authors: Bhimani, Divyang G., Carles, Rémi
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2020
Springer
Springer Nature B.V
Springer Verlag
Subjects:
Abstract Harmonic Analysis
Analysis of PDEs
Approximations and Expansions
Fourier Analysis
Functional Analysis
Geometrical optics
Inflation (Finance)
Mathematical analysis
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Modulation
Nonlinear equations
Nonlinearity
Partial Differential Equations
Regularity
Schrodinger equation
Signal,Image and Speech Processing
Fourier–Lebesgue spaces
Ill-posedness
35Q55
35A01 (secondary)
Nonlinear Schrödinger equations
Modulation spaces
42B35 (primary)
ISSN:1069-5869, 1531-5851
Online Access:Get full text
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Summary:We consider nonlinear Schrödinger equations in Fourier–Lebesgue and modulation spaces involving negative regularity. The equations are posed on the whole space, and involve a smooth power nonlinearity. We prove two types of norm inflation results. We first establish norm inflation results below the expected critical regularities. We then prove norm inflation with infinite loss of regularity under less general assumptions. To do so, we recast the theory of multiphase weakly nonlinear geometric optics for nonlinear Schrödinger equations in a general abstract functional setting.
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ISSN:1069-5869
1531-5851
DOI:10.1007/s00041-020-09788-w