Pathological Set of Initial Data for Scaling-Supercritical Nonlinear Schrödinger Equations-Reference-Cited by-同舟云学术

Pathological Set of Initial Data for Scaling-Supercritical Nonlinear Schrödinger Equations

Published:2022-07-20 Issue: Volume: Page:
ISSN:1073-7928
Container-title:International Mathematics Research Notices
language:en
Short-container-title:

Author:

Camps Nicolas¹,Gassot Louise²

Affiliation:

1. Université Paris-Saclay, Laboratoire de Mathématiques d’Orsay , UMR 8628 du CNRS, Bâtiment 307, 91405 Orsay Cedex, France

2. ICERM, Brown University, 121 South Main Street , Providence, RI 02903, USA

Abstract

Abstract The purpose of this work is to evidence a pathological set of initial data for which the regularized solutions by convolution experience a norm-inflation mechanism, in arbitrarily short time. The result is in the spirit of the construction from Sun and Tzvetkov, where the pathological set contains a superposition of profiles that concentrate at different points. Thanks to finite propagation speed of the wave equation, and given a certain time, at most one profile exhibits significant growth. However, for Schrödinger-type equations, we cannot preclude the profiles from interacting between each other. Instead, we propose a method that exploits the regularizing effect of the approximate identity, which, at a given scale, rules out the norm inflation of the profiles that are concentrated at smaller scales.

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

Link

https://academic.oup.com/imrn/advance-article-pdf/doi/10.1093/imrn/rnac194/45033321/rnac194.pdf

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