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On the global well-posedness for the periodic quintic nonlinear Schrödinger equation

Xueying Yu, Haitian Yue

Abstract

In this paper, we consider the initial value problem for the quintic, defocusing nonlinear Schrödinger equation on $\Bbb T^2$ with general data in the critical Sobolev space $H^{\frac{1}{2}} (\Bbb T^2)$. We show that if a solution remains bounded in $H^{\frac{1}{2}} (\Bbb T^2)$ in its maximal interval of existence, then the solution is globally well-posed in $\Bbb T^2$.

On the global well-posedness for the periodic quintic nonlinear Schrödinger equation

Abstract

In this paper, we consider the initial value problem for the quintic, defocusing nonlinear Schrödinger equation on with general data in the critical Sobolev space . We show that if a solution remains bounded in in its maximal interval of existence, then the solution is globally well-posed in .

Paper Structure

This paper contains 24 sections, 29 theorems, 235 equations.

Table of Contents

  1. Introduction
  2. Setup of NLS
  3. History and related works
  4. Well-posedness results in $\mathbb{R}^d$
  5. Well-posedness results in $\mathbb{T}^d$ and more
  6. Main result
  7. Outline of the proof
  8. Organization of the paper
  9. Preliminaries
  10. Notations
  11. Definition of solutions
  12. Atomic spaces
  13. Local well-posedness and stability
  14. Strichartz estimates
  15. Trilinear estimates
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $u_0 \in H^{\frac{1}{2}}(\mathbb{T}^2)$ and $u : I \times \mathbb{R}^2 \to \mathbb{C}$ be a maximal-lifespan solution to NLS with initial data $u_0$ such that for any $T>0$, Then the solution is globally well-posed. In addition, the mapping $u_0 \mapsto u$ extends to a continuous mapping from $H^{\frac{1}{2}} (\mathbb{T}^2)$ to the $X^{\frac{1}{2}} ([-T, T])$ spaceNote that the space $X^{\fr

Theorems & Definitions (66)

  • Theorem 1.1: Global well-posedness of \ref{['NLS']}
  • Definition 2.1: Solutions
  • Definition 2.2: $U_{\Delta}^p$-spaces, Definitions 2.1, 2.15 in HHK
  • Definition 2.3: $V_{\Delta}^p$-spaces, Definitions 2.3, 2.15 in HHK
  • Proposition 2.4: Embedding, Proposition 2.2, Proposition 2.4 and Corollary 2.6 in HHK
  • Proposition 2.5: Duality, Theorem 2.8 in HHK
  • Proposition 2.6: Transfer Principle, Proposition 2.19 in HHK
  • Definition 2.7: $X^s$ and $Y^s$ spaces
  • Proposition 2.8
  • Definition 2.9: $Z$-norm
  • ...and 56 more