Scattering below the ground state for the three-dimensional focusing energy-critical NLS
Abstract
We prove global well-posedness and scattering for the three-dimensional focusing energy-critical nonlinear Schrödinger equation with initial data below the ground-state energy and gradient thresholds. This extends the seminal result of Kenig and Merle (Invent. Math., 2006) to general data in Ḣ1(ℝ3). Classical concentration compactness and the exclusion of rapid frequency cascades reduce the problem to the quasi-soliton scenario. The main ingredient is a frequency-localized interaction Morawetz estimate based on an explicit convex weight whose time-dependent radius is determined implicitly by the mass density of the high-frequency component. The ground-state gap controls the focusing contribution with constants independent of the mass of this component. The radius is chosen to retain quartic spacetime control, while the continuity equation expresses the moving-radius contribution as a nonnegative square, up to frequency-truncation errors. We absorb these errors by closing the Morawetz estimate together with an auxiliary spacetime bound for the nonlinearity. The resulting quartic bound excludes the quasi-soliton scenario and completes the proof.
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BibTeX
@misc{su2026scattering,
title = {{Scattering below the ground state for the three-dimensional focusing energy-critical NLS}},
author = {Qingtang Su and Zehua Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.30228},
eprint = {2609.30228},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30228},
}
RIS
TY - UNPB
TI - Scattering below the ground state for the three-dimensional focusing energy-critical NLS
AU - Qingtang Su
AU - Zehua Zhao
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2609.30228
UR - https://arxiv.org/abs/2609.30228
AB - We prove global well-posedness and scattering for the three-dimensional focusing energy-critical nonlinear Schrödinger equation with initial data below the ground-state energy and gradient thresholds. This extends the seminal result of Kenig and Merle (Invent. Math., 2006) to general data in Ḣ^1(ℝ^3). Classical concentration compactness and the exclusion of rapid frequency cascades reduce the problem to the quasi-soliton scenario. The main ingredient is a frequency-localized interaction Morawetz estimate based on an explicit convex weight whose time-dependent radius is determined implicitly by the mass density of the high-frequency component. The ground-state gap controls the focusing contribution with constants independent of the mass of this component. The radius is chosen to retain quartic spacetime control, while the continuity equation expresses the moving-radius contribution as a nonnegative square, up to frequency-truncation errors. We absorb these errors by closing the Morawetz estimate together with an auxiliary spacetime bound for the nonlinearity. The resulting quartic bound excludes the quasi-soliton scenario and completes the proof.
ER -