Global well-posedness and scattering for the mass critical nonlinear\n Schr "odinger equation with mass below the mass of the ground state

2011/04/06 by Benjamin Dodson, Dodson, Benjamin · 5 citations
Mathematics · #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1104.1114

openalex publication_date 2011/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we prove that the focusing, d-dimensional mass critical\nnonlinear Schr "odinger initial value problem is globally well-posed and\nscattering for u₀ \∈ L²(\R^d), \‖ u₀\n\‖_L²(\R^d) < \‖ Q \‖_L²(\R^d), where Q is the\nground state, and d \≥ 1. We first establish an interaction Morawetz\nestimate that is positive definite when \‖ u₀ \‖_L²(\R^d) <\n\‖ Q \‖_L²(\R^d), and has the appropriate scaling. Next, we\nwill prove a frequency localized interaction Morawetz estimate similar to the\nestimates made in citeD2, citeD3, citeD4. See also citeCKSTT4 for\nthe energy critical case. Since we are considering an L² - critical\ninitial value problem we will localize to low frequencies.\n

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