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Norm inflation for quadratic derivative fractional nonlinear Schrödinger equations

Toshiki Kondo, Mamoru Okamoto

TL;DR

This work determines sharp Sobolev-regularity thresholds for the Cauchy problem $\partial_t u + i D^\alpha u = u D^\beta u$ for quadratic derivative fractional NLS on $\mathbb{R}$ and $\mathbb{T}$. By leveraging the global well-posedness framework of ${\mathcal{F}}^{-1} \mathcal{X}$ from NaWa25, the authors expand solutions into iterates and identify the dominant terms that drive norm inflation. They prove ill-posedness on the real line whenever $\alpha>0$ and $\beta>\max\left(\tfrac{\alpha-1}{2},0\right)$, with infinite loss of regularity, and show ill-posedness on the torus for all $\alpha,\beta>0$, implying no local smoothing in the periodic setting. The results provide optimal thresholds and unify the real-line and torus behaviors within a distribution-space framework, clarifying how derivative losses interact with dispersion in quadratic derivative NLS.

Abstract

We consider the Cauchy problem for quadratic derivative fractional nonlinear Schrödinger equations on $\mathbb{R}$ or $\mathbb{T}$. We determine the sharp exponents of the fractional derivatives for which the Cauchy problem is well-posed in the Sobolev space. Thanks to the global well-posedness result established by Nakanishi and Wang (2025), we can expand the solution as a sum of iterated terms. By deriving estimates for each iterated term, we establish norm inflation with infinite loss of regularity, which in particular implies ill-posedness.

Norm inflation for quadratic derivative fractional nonlinear Schrödinger equations

TL;DR

This work determines sharp Sobolev-regularity thresholds for the Cauchy problem for quadratic derivative fractional NLS on and . By leveraging the global well-posedness framework of from NaWa25, the authors expand solutions into iterates and identify the dominant terms that drive norm inflation. They prove ill-posedness on the real line whenever and , with infinite loss of regularity, and show ill-posedness on the torus for all , implying no local smoothing in the periodic setting. The results provide optimal thresholds and unify the real-line and torus behaviors within a distribution-space framework, clarifying how derivative losses interact with dispersion in quadratic derivative NLS.

Abstract

We consider the Cauchy problem for quadratic derivative fractional nonlinear Schrödinger equations on or . We determine the sharp exponents of the fractional derivatives for which the Cauchy problem is well-posed in the Sobolev space. Thanks to the global well-posedness result established by Nakanishi and Wang (2025), we can expand the solution as a sum of iterated terms. By deriving estimates for each iterated term, we establish norm inflation with infinite loss of regularity, which in particular implies ill-posedness.
Paper Structure (11 sections, 18 theorems, 190 equations)

This paper contains 11 sections, 18 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. On the real-line
  3. On the torus
  4. Notations
  5. Organization of the paper
  6. Global well-posedness in distributions on the Fourier half-space
  7. The space of distributions supported on the Fourier half-space
  8. Global well-posedness
  9. Ill-posedness on the real-line
  10. Ill-posedness on the torus
  11. Well-posedness on the real-line

Key Result

Theorem 1.2

When $\mathcal{M} = \mathbb{R}$, let $\alpha, \beta \in \mathbb{R}$ satisfy $\alpha>0$ and Let $s, \sigma \in \mathbb{R}$. Then, for any $0< \varepsilon \ll 1$, there exist an initial datum $\phi \in H^s(\mathbb{R})$ with $\| \phi \|_{H^s} < \varepsilon$ and $\mathop{\mathrm{supp}}\limits \widehat{\phi} \subset [0, \infty)$ and a time $T \in (0,\varepsilon)$ such that the corresponding so

Theorems & Definitions (40)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • ...and 30 more