Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global solutions for cubic quasilinear Schroedinger flows in two and higher dimensions

Mihaela Ifrim, Daniel Tataru

Abstract

In recent work the authors proposed a broad global well-posedness conjecture for cubic defocusing dispersive equations in one space dimension, and then proved this conjecture in two cases, namely for one dimensional semilinear and quasilinear Schrödinger flows. Inspired by the circle of ideas developed in the proof of the above conjecture, in this paper we expand the reach of these methods to higher dimensional quasilinear cubic Schrödinger flows. 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 (MMT) for initial data in Sobolev spaces. The outcomes of this work are (i) a new, potentially sharp local well-posedness result in low regularity Sobolev spaces, one derivative below MMT and just one half derivative above scaling, (ii) a small data global well-posedness and scattering result at the same regularity level, the first result of its kind at least in two space dimensions, and (iii) a new way to think about this class of problems, which, we believe, will become the standard approach in the future.

Global solutions for cubic quasilinear Schroedinger flows in two and higher dimensions

Abstract

In recent work the authors proposed a broad global well-posedness conjecture for cubic defocusing dispersive equations in one space dimension, and then proved this conjecture in two cases, namely for one dimensional semilinear and quasilinear Schrödinger flows. Inspired by the circle of ideas developed in the proof of the above conjecture, in this paper we expand the reach of these methods to higher dimensional quasilinear cubic Schrödinger flows. 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 (MMT) for initial data in Sobolev spaces. The outcomes of this work are (i) a new, potentially sharp local well-posedness result in low regularity Sobolev spaces, one derivative below MMT and just one half derivative above scaling, (ii) a small data global well-posedness and scattering result at the same regularity level, the first result of its kind at least in two space dimensions, and (iii) a new way to think about this class of problems, which, we believe, will become the standard approach in the future.
Paper Structure (35 sections, 31 theorems, 321 equations)

This paper contains 35 sections, 31 theorems, 321 equations.

Table of Contents

  1. Introduction
  2. The global well-posedness conjectures
  3. Quasilinear Schrödinger flows: local solutions
  4. Quasilinear Schrödinger flows: global solutions
  5. Outline of the paper
  6. Acknowledgements
  7. Notations and preliminaries
  8. The Littlewood-Paley decomposition
  9. Strichartz and bilinear $L^2_{t,x}$ bounds
  10. Resonant analysis
  11. Translation invariant multilinear forms
  12. A frequency envelope formulation of the results
  13. Frequency envelopes
  14. The frequency envelope form of the results
  15. The bootstrap hypotheses
  16. ...and 20 more sections

Key Result

Theorem 1

The nD cubic problem qnls is locally well-posed for nontrapping data in $H^s$ for $s > \frac{n+3}{2}$, and the cubic problem dqnls is locally well-posed in $H^s$ for $s > \frac{n+5}{2}$.

Theorems & Definitions (57)

  • Conjecture 1: Non-localized data defocusing GWP conjecture
  • Conjecture 2: Non-localized data long-time well-posedness conjecture
  • Conjecture 3: 2D non-localized data GWP conjecture
  • Definition 1.1
  • Theorem 1: cubic nonlinearities MMT2MMT3
  • Theorem 2: 1D cubic nonlinearities IT-qnls
  • Theorem 3
  • Theorem 4
  • Conjecture 4
  • Conjecture 5
  • ...and 47 more