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

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.

Refined probabilistic local well-posedness for a cubic Schrödinger half-wave equation

“…It remains to study the time evolution of ρ εn k * f , by comparing it to the ODE profile ρ εn k * v 0,k only. An adaptation of the proof of Proposition 2.6 in [12] leads to the following result. Let u εn k be the maximal solution of (NLS-HW) with initial data ρ εn k * f in L 2…”

“…In order to define a dense subset S of the pathological set One can see that S defines a dense subset of H s . We make use of the following precise upper bounds in H m , m ≥ 0, of the initial data regularized by convolution, see Lemma 2.4 in [12]. .…”

“…has also been investigated in [31]. Using the method of Sun and Tzvetkov [35] adapted to Schrödinger-type equations in [12], we make precise this norm inflation result in the context of probabilistic well-posedness. We fix ρ ∈ C ∞ c (R 2 ), valued in [0, 1], such that R 2 ρ(x) dx = 1 and ρ vanishes for x 2 + y 2 ≥ 1 10 4 .…”

“…Since the argument is the same as in [12], we only make a sketch of the proof adapted to the anisotropy. We construct unstable profiles with growing H s norm based on the bounded time-periodic solution V (t) = e it to the ODE…”