Pathological set of initial data for scaling-supercritical nonlinear Schr\"odinger equations

TL;DR

This paper constructs a pathological set of initial data for supercritical nonlinear Schrödinger equations where solutions exhibit rapid norm inflation, highlighting potential instabilities in the regularized solution process.

Contribution

It introduces a novel method to analyze interactions of concentrated profiles, demonstrating norm inflation in supercritical Schrödinger equations despite regularization effects.

Findings

Identifies a pathological initial data set causing norm inflation

Shows that profile interactions can lead to instability

Proposes a method leveraging regularization to control profile interactions

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 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 Schr\"odinger-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.