Fractional nonlinear Schrödinger and Hartree equations in modulation spaces

TL;DR

This work establishes local and global well-posedness for fractional nonlinear Schrödinger and Hartree equations with radial data in modulation spaces $M^{p,p'}_{rad}$ for fractional dispersion $β\in(\frac{2n}{2n-1},2)$ and spatial dimension $n\ge 2$. The authors adapt Bourgain’s high–low frequency decomposition to the fractional setting and exploit radial Strichartz estimates to control nonlinear and nonlocal terms, achieving global well-posedness for large data in modulation spaces and extending known results beyond Sobolev spaces. The results cover both power-type and Hartree nonlinearities, with sharp global bounds in $M^{p,p'}_{rad}$ and $M^{s,s'}_{rad}$ for appropriate ranges of the exponents and regularity. The radial symmetry is crucial for the Strichartz framework and enables global control without conservation laws in modulation spaces, yielding novel large-data well-posedness in the fractional dispersion regime and clarifying thresholds in modulation-space scales.

Abstract

We establish global well-posedness for the mass subcritical nonlinear fractional Schrödinger equation $$iu_t - (-Δ)^\fracβ{2} u+F(u)=0$$ having radial initial data in modulation spaces $M^{p,\frac{p}{p-1}}(\mathbb R^n)$ for $n \geq 2, p>2$ and $p$ sufficiently close to $2.$ The nonlinearity $F(u)$ is either of power-type $F(u)=\pm (|u|^αu)\; (0<α<2β/ n)$ or Hartree-type $(|x|^{-ν} \ast |u|^{2})u \; (0<ν<\min\{β,n\}).$ Our order of dispersion $β$ lies in $(2n/ (2n-1), 2).$

Theorems & Definitions (36)

• Theorem 1.1: Local well-posedness
• Theorem 1.2: global well-posedness
• Theorem 1.3: local well-posedness
• Theorem 1.4: global well-posedness
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Definition 2.1
• Proposition 2.1: Strichartz estimates for fractional Schrödinger equation
• Lemma 2.2: see e.g. Lemma 3.9 in leonidthesis