Uniform local smoothing for the derivative NLS and the soliton mass threshold
Abstract
Harrop-Griffiths, Killip, Ntekoume and Vişan proved that the derivative nonlinear Schrödinger equation (DNLS) is globally well-posed in L2(ℝ). A key ingredient of their proof is a local smoothing estimate. It holds on L2-equicontinuous sets of initial data, but it cannot hold on L2-balls, because rescaled stationary solitons, all of mass 2π, violate it. This raises a natural question: for which masses m does local smoothing hold uniformly on the ball {M(q0)≤ m}? We prove three results. First, we give a quantitative version of the local smoothing estimate in which the data enter only through the mass and an explicit frequency scale κ(q0). On time intervals of length 2T the bound grows like 1+κ(q0)2T, and the stationary soliton shows that the power of κ cannot be lowered below one. Second, we prove uniform local smoothing on balls of small mass, together with a scale-invariant Morrey-type form and a localized L4t,x bound. Third, we analyse the soliton family from both sides. The uniform constant must blow up at least like (2π−m)−1 as m↑ 2π, while every soliton of mass m<2π satisfies local smoothing with a constant O((2π−m)−1). These facts single out the stationary-soliton mass 2π, which is also the Hayashi--Ozawa coercivity threshold, as the natural candidate for the sharp threshold. We formulate this as a conjecture, give a spectral reformulation in terms of the Kaup--Newell scattering data, and discuss the obstacles to a proof.
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BibTeX
@misc{majdoub2026uniform,
title = {{Uniform local smoothing for the derivative NLS and the soliton mass threshold}},
author = {Mohamed Majdoub},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05330},
eprint = {2611.05330},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05330},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniform local smoothing for the derivative NLS and the soliton mass threshold
AU - Mohamed Majdoub
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05330
UR - https://arxiv.org/abs/2611.05330
AB - Harrop-Griffiths, Killip, Ntekoume and Vişan proved that the derivative nonlinear Schrödinger equation (DNLS) is globally well-posed in L^2(ℝ). A key ingredient of their proof is a local smoothing estimate. It holds on L^2-equicontinuous sets of initial data, but it cannot hold on L^2-balls, because rescaled stationary solitons, all of mass 2π, violate it. This raises a natural question: for which masses m does local smoothing hold uniformly on the ball {M(q_0)≤ m}? We prove three results. First, we give a quantitative version of the local smoothing estimate in which the data enter only through the mass and an explicit frequency scale κ(q_0). On time intervals of length 2T the bound grows like 1+κ(q_0)^2T, and the stationary soliton shows that the power of κ cannot be lowered below one. Second, we prove uniform local smoothing on balls of small mass, together with a scale-invariant Morrey-type form and a localized L^4_t,x bound. Third, we analyse the soliton family from both sides. The uniform constant must blow up at least like (2π−m)^−1 as m↑ 2π, while every soliton of mass m<2π satisfies local smoothing with a constant O((2π−m)^−1). These facts single out the stationary-soliton mass 2π, which is also the Hayashi--Ozawa coercivity threshold, as the natural candidate for the sharp threshold. We formulate this as a conjecture, give a spectral reformulation in terms of the Kaup--Newell scattering data, and discuss the obstacles to a proof.
ER -