Local well-posedness for cubic fractional Schrödinger equations with derivatives on the right-hand side

TL;DR

This work establishes local well-posedness for a cubic cubic-type fractional Schrödinger equation with derivatives on the right-hand side for $s\in(\tfrac12,1)$ and $n\ge 4$, under small $\dot{H}^{\frac{n-2s}{2}}$ data. The authors develop a resolution-space framework—$X_k$, $Y_k^e$, $Z_k$ and global $F^\sigma$, $N^\sigma$ spaces—and derive cone-local smoothing and maximal estimates to handle the derivative nonlinearity. Central to the approach are linear and trilinear estimates that control the nonlinear term in these spaces, enabling a contraction mapping argument that yields a unique, continuously dependent local solution with energy-type bounds. The results illuminate a model for $s$-Schrödinger maps and extend local well-posedness techniques from Schrödinger maps and half-wave maps to the fractional regime $\tfrac12< s<1$. The framework provides a robust toolkit (cone estimates, smoothing, and dyadic analysis) for analyzing similar dispersive PDEs with derivative nonlinearities.

Abstract

For $s \in (\frac{1}{2},1]$ we investigate well-posedness of the equation \[ \left ( i \partial_t + (-Δ)^{s} \right ) u = \left (|D|^{1-2s} |u|^2 \right)\ |D|^{2s-1} u \] under small initial data $\|u(0)\|_{H^{\frac{n-2s}{2}}(\mathbb{R}^n)} \ll 1$. This equation is a model equation for for $s$-Schrödinger map equation \[ \partial_t ψ= ψ\wedge (-Δ)^s ψ: \quad ψ: \mathbb{R}^n \times \mathbb{R} \to \mathbb{S}^{2}, \]

Theorems & Definitions (52)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• proof