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Well-posedness in the full scaling-subcritical range for a class of nonlocal NLS on the line

Sonae Hadama

Abstract

In this paper, we study a class of one-dimensional nonlocal nonlinear Schrödinger equations on the line with nonlinearity given by a Fourier multiplier whose symbol has subcritical high-frequency growth. In terms of symbol order, this class is intermediate between the cubic nonlinear Schrödinger equation and the Calogero--Moser derivative nonlinear Schrdöinger equation. We prove local well-posedness in $L^2(\mathbb{R})$ throughout the full scaling-subcritical range. Due to derivative loss, the standard Duhamel integral is not directly meaningful for rough data. To avoid this problem, we first construct the propagator $S_V$ for rough time-dependent potentials $V$, and then prove an Ozawa-Tsutsumi type bilinear Strichartz estimate for the perturbed flow $S_V$. These linear theories yield a concrete construction of rough solutions without using any equation-specific algebraic structure. For real-valued symbols, mass is conserved, and the local solutions are therefore global.

Well-posedness in the full scaling-subcritical range for a class of nonlocal NLS on the line

Abstract

In this paper, we study a class of one-dimensional nonlocal nonlinear Schrödinger equations on the line with nonlinearity given by a Fourier multiplier whose symbol has subcritical high-frequency growth. In terms of symbol order, this class is intermediate between the cubic nonlinear Schrödinger equation and the Calogero--Moser derivative nonlinear Schrdöinger equation. We prove local well-posedness in throughout the full scaling-subcritical range. Due to derivative loss, the standard Duhamel integral is not directly meaningful for rough data. To avoid this problem, we first construct the propagator for rough time-dependent potentials , and then prove an Ozawa-Tsutsumi type bilinear Strichartz estimate for the perturbed flow . These linear theories yield a concrete construction of rough solutions without using any equation-specific algebraic structure. For real-valued symbols, mass is conserved, and the local solutions are therefore global.

Paper Structure

This paper contains 15 sections, 14 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Novelty and idea of the proof
  4. Novelty
  5. Idea of the proof
  6. Construction of $S_V$
  7. Dyson series expansion
  8. Short-time construction under a smallness assumption
  9. Basic properties
  10. Extension to general potentials
  11. Bilinear Strichartz estimates for $S_V$
  12. Proof of the well-posedness result
  13. Local well-posedness
  14. Global well-posedness
  15. Christ--Kiselev type lemmas

Key Result

Theorem 1.1

Assume that $0< \delta <1$ and $a:\mathbb{R}\to \mathbb{C}$ satisfies eq:assmption for m. Then, for any $\phi\in L^2(\mathbb{R})$, there exist $T=T(a,\delta,\|\phi\|_{L^2_x(\mathbb{R})})>0$ and a unique solution $u(t)\in C([0,T];L^2_x(\mathbb{R}))$ to eq:NLS with initial condition $u(0)=\phi$ such t for all $t\in [0,\infty)$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Definition 2.4: Definition of $S_V$ with a smallness assumption
  • ...and 26 more