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Sharp well-posedness and ill-posedness results for the inhomogeneous NLS equation

Luccas Campos, Simão Correia, Luiz Gustavo Farah

Abstract

We consider the initial value problem associated to the inhomogeneous nonlinear Schrö\-din\-ger equation, \begin{equation} iu_t + Δu +μ|x|^{-b}|u|^αu=0, \quad u_0\in H^s(\mathbb R^N) \text{ or } u_0 \in\dot H ^s(\mathbb R^N), \end{equation} with $μ=\pm 1$, $b > 0$, $s\geq 0$ and $0 < α\leq \frac{4-2b}{N-2s}$. By means of an adapted version of the fractional Leibniz rule, we prove new local well-posedness results in Sobolev spaces for a large range of parameters. We also prove an ill-posedness result for this equation, through a delicate analysis of the associated Duhamel operator.

Sharp well-posedness and ill-posedness results for the inhomogeneous NLS equation

Abstract

We consider the initial value problem associated to the inhomogeneous nonlinear Schrö\-din\-ger equation, \begin{equation} iu_t + Δu +μ|x|^{-b}|u|^αu=0, \quad u_0\in H^s(\mathbb R^N) \text{ or } u_0 \in\dot H ^s(\mathbb R^N), \end{equation} with , , and . By means of an adapted version of the fractional Leibniz rule, we prove new local well-posedness results in Sobolev spaces for a large range of parameters. We also prove an ill-posedness result for this equation, through a delicate analysis of the associated Duhamel operator.
Paper Structure (12 sections, 23 theorems, 183 equations)

This paper contains 12 sections, 23 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Setting and motivation
  3. Statement of the results
  4. Preliminaries
  5. Notation
  6. Linear estimates
  7. Fractional chain and Leibniz rules
  8. Nonlinear estimates
  9. Well-posedness
  10. Well-posedness in inhomogeneous Sobolev spaces
  11. Well-posedness in homogeneous Sobolev spaces
  12. Ill-posedness

Key Result

Theorem 1.1

Let $N \geq 1$, $s\geq 0$, $\alpha>0$ and $0 < b <\min\{2,N-s,\frac{N}{2}+1-s \}$. Moreover, if $\alpha$ is not an even integer, assume additionally that $s<\alpha+1$. In both cases, if $s\le 1$, given $\delta>0$, there exists $\epsilon>0$ such that, for any $v_0\in H^s$ with $\|u_0-v_0\|_{H^s}<\epsilon$, the solution $v$ with initial data $v_0$ is defined on $[0,T]$ and satisfies

Theorems & Definitions (51)

  • Theorem 1.1: Well-posedness in ${H}^s$
  • Remark 1.2
  • Theorem 1.3: Well-posedness in $\dot{H}^s$
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7: Ill-posedness
  • Remark 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 41 more