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 $L^2(\mathbb{R})$. 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 $\{M(q_0)\le m\}$? 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 $m\uparrow 2π$, 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.