Global solutions for 1D cubic dispersive equations, Part III: the quasilinear Schrödinger flow
Mihaela Ifrim, Daniel Tataru
TL;DR
This paper proves sharp local well-posedness for 1D quasilinear Schrödinger flows with cubic nonlinearities at minimal Sobolev regularity and establishes global well-posedness with scattering for small data in the defocusing, phase-rotation-symmetric, conservative setting. The authors develop a sophisticated analytic framework combining paradifferential calculus, frequency envelopes, density-flux identities, and interaction Morawetz estimates to control both local and long-time behavior without localization of data. Key contributions include a frequency-envelope formulation of the results, a detailed para-differential and resonant expansion that isolates nonperturbative interactions, and novel Strichartz estimates at low regularity alongside robust Morawetz-type bounds. The results yield the first global well-posedness and scattering results for a 1D quasilinear dispersive flow, with quantitative, frequency-localized energy and dispersive bounds that scale to the ε^{-8} long-time regime in the conservative case and to global times in the defocusing case. Overall, the work bridges local theory and global dispersive behavior for 1D quasilinear Schrödinger equations, advancing understanding of how nonlinear and dispersive effects interact in low dimensions.
Abstract
The first target of this article is the local well-posedness question for 1D quasilinear Schrödinger equations with cubic nonlinearities. The study of this class of problems, in all dimensions, was initiated in pioneering work of Kenig-Ponce-Vega for localized initial data, and then continued by Marzuola-Metcalfe-Tataru for initial data in Sobolev spaces. Our objective here is to fully redevelop the study of this problem in the 1D case, and to prove a \emph{sharp local well-posedness} result. The second goal of this article is to consider the long time/global existence of solutions for the same problem. This is motivated by a broad conjecture formulated by the authors in earlier work, which reads as follows: ``\emph{Cubic defocusing dispersive one dimensional flows with small initial data have global dispersive solutions}''; the conjecture was initially proved for a well chosen semilinear model of Schrödinger type. Our work here establishes the above conjecture for 1D quasilinear Schrödinger flows. Precisely, we show that if the problem has \emph{phase rotation symmetry} and is \emph{conservative and defocusing}, then small data in Sobolev spaces yields global, scattering solutions. This is the first result of this type for 1D quasilinear dispersive flows. Furthermore, we prove it at the minimal Sobolev regularity in our local well-posedness result. The defocusing condition is essential in our global result. Without it, the authors have conjectured that \emph{small, $ε$ size data yields long time solutions on the $ε^{-8}$ time-scale}. A third goal of this paper is to also prove this second conjecture for 1D quasilinear Schrödinger flows.