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Global well-posedness and scattering of the defocusing energy-critical inhomogeneous nonlinear Schrödinger equation with radial data

Dongjin Park

Abstract

We consider the defocusing energy-critical inhomogeneous nonlinear Schrödinger equation (INLS) $iu_t + Δu = |x|^{-b}|u|^{k}u$ in $\mathbb{R} \times \mathbb{R}^{n}$ where $n \geq 3$, $0<b<\min(2, n/2)$, and $k=(4-2b)/(n-2)$. We show that for every spherically symmetric initial data $φ\in H^1(\mathbb{R}^n)$, or preferably $\dot{H}^1(\mathbb{R}^n)$, the solution is globally well-posed and scatters for every such $n$ and $b$ except for $n=4$ with $1\leq b<2$ and $n=5$ with $1/2\leq b\leq 5/4$. We mainly apply the arguments of Tao (2005), but inspired by the work of Aloui and Tayachi (2021), we utilize Lorentz spaces to define spacetime norms. This method is distinct from the widespread concentration compactness principle and establishes a quantitative bound for the solution's spacetime norm. The bound has an exponential form $C\exp(CE[φ]^C)$ in terms of the energy $E[φ]$, similar to Tao's work.

Global well-posedness and scattering of the defocusing energy-critical inhomogeneous nonlinear Schrödinger equation with radial data

Abstract

We consider the defocusing energy-critical inhomogeneous nonlinear Schrödinger equation (INLS) in where , , and . We show that for every spherically symmetric initial data , or preferably , the solution is globally well-posed and scatters for every such and except for with and with . We mainly apply the arguments of Tao (2005), but inspired by the work of Aloui and Tayachi (2021), we utilize Lorentz spaces to define spacetime norms. This method is distinct from the widespread concentration compactness principle and establishes a quantitative bound for the solution's spacetime norm. The bound has an exponential form in terms of the energy , similar to Tao's work.
Paper Structure (12 sections, 18 theorems, 136 equations)

This paper contains 12 sections, 18 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Local Well-posedness and Other Basic Estimates
  3. Definitions and Properties of Sobolev-Lorentz Spaces
  4. Some Local Well-posedness Results
  5. Morawetz Inequality for INLS
  6. Local Mass Conservation
  7. Proof of Theorem \ref{['thrm:main']}
  8. The Main Road Map
  9. Proof of Proposition \ref{['prop:3.3-locmass']}
  10. Proof of Lemma \ref{['lem:3.8-v-holder-est']}
  11. Proof of Lemma \ref{['lem:3.2']} for Higher Dimensions
  12. Proof of Lemma \ref{['lem:4.1']}

Key Result

Theorem 1.1

Let $n\geq 3$ be an integer and $b>0$ obey the following range restriction. Let $[T_-, T_+]$ be a compact interval, and let $u$ be the unique spherically symmetric solution of inls associated with the initial data $\phi \in H^1(\mathbb{R}^n)$ and lying in the solution space where $p_* = 2(n+2)/(n-2+2b)$ and $q_* = 2n(n+2)/(n^2+4-4b)$. Then we have where $E = E[\phi]$ is the solution's energy an

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Remark
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark
  • Proposition 2.1: Hölder's inequality; Lemarie02, ONeil63
  • Proposition 2.2: Sobolev embeddings; Lemarie02
  • Proposition 2.3: Strichartz estimates; KeelTao
  • ...and 20 more