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Global well-posedness for the 1D cubic nonlinear Schrödinger equation in $L^p,\,p>2$

Ryosuke Hyakuna

Abstract

In this paper, we show that the one dimensional cubic nonlinear Schrödinger equation is globally well posed in $L^p$ for $2\le p <13/6$. In particular, we prove that the global solution enjoys the persistence property for a twisted variable at any time, which implies the result is a natural exetension of the classical global well-posedness in $L^2$ to $L^p$. The proof exploits the data-decomposition argument originally developed by Vargas-Vega in the functional framework introduced by Zhou.

Global well-posedness for the 1D cubic nonlinear Schrödinger equation in $L^p,\,p>2$

Abstract

In this paper, we show that the one dimensional cubic nonlinear Schrödinger equation is globally well posed in for . In particular, we prove that the global solution enjoys the persistence property for a twisted variable at any time, which implies the result is a natural exetension of the classical global well-posedness in to . The proof exploits the data-decomposition argument originally developed by Vargas-Vega in the functional framework introduced by Zhou.

Paper Structure

This paper contains 8 sections, 14 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Function spaces and key estimates
  3. Function spaces
  4. Nonlinear estimates
  5. Proof of Theorem \ref{['Maintheorem']}
  6. A key global existence theorem
  7. Proof of Theorem \ref{['keyGWP']}
  8. Proof of Theorem \ref{['Maintheorem']}

Key Result

Theorem 1.2

Let $2 \le p < 13/6$. Then (NLS) is globally well posed in $L^p$ in the sense of Definition LWPdef. More precisely, for any $\phi \in L^p$ there exists a unique global solution $u\in C_{\mathfrak{S}}(\mathbb{R}\,; L^p(\mathbb{R}))$ of (NLS) such that for all $T>0$, where $\epsilon >0$ is a sufficiently small constant determined depending only on $\epsilon$. Moreover, for any $r \in [3,6]$.

Theorems & Definitions (27)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Corollary 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 17 more