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Dispersive decay for the mass-critical nonlinear Schrödinger equation

Chenjie Fan, Rowan Killip, Monica Visan, Zehua Zhao

Abstract

We prove dispersive decay, pointwise in time, for solutions to the mass-critical nonlinear Schrödinger equation in spatial dimensions $d=1,2,3$.

Dispersive decay for the mass-critical nonlinear Schrödinger equation

Abstract

We prove dispersive decay, pointwise in time, for solutions to the mass-critical nonlinear Schrödinger equation in spatial dimensions .
Paper Structure (5 sections, 9 theorems, 81 equations)

This paper contains 5 sections, 9 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Notation and useful lemmas
  3. Proof of Theorem \ref{['T:2d']}
  4. Proof of Theorem \ref{['T:1d']}
  5. Proof of Theorem \ref{['T:3d']}

Key Result

Theorem 1.1

Fix $d\geq 1$ and let $u_0\in L^2(\mathbb{R}^d)$. In the focusing case assume also that $M(u_0)<M(Q)$. Then there exists a unique global solution $u\in (C_tL_x^2\cap L_{t,x}^{2(d+2)/d})(\mathbb{R}\times\mathbb{R}^d)$ to NLS and Moreover, there exist asymptotic states $u_\pm\in L^2(\mathbb{R}^d)$ such that

Theorems & Definitions (13)

  • Theorem 1.1: Dodson
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: Lorentz spaces
  • Lemma 2.2: Hölder inequality in Lorentz spaces
  • Proposition 2.3: Lorentz--Strichartz estimates
  • Lemma 2.4: Lorentz spacetime bounds $d=1$
  • proof
  • Lemma 2.5: Lorentz spacetime bounds $d=2$
  • ...and 3 more