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

TL;DR

This paper establishes local well-posedness for a class of one-dimensional nonlocal nonlinear Schrödinger equations with subcritical high-frequency growth, using novel linear theory constructions to handle rough data.

Contribution

It introduces a new approach to prove well-posedness for nonlocal NLS with derivative loss by constructing a propagator for rough potentials and establishing bilinear Strichartz estimates.

Findings

Proved local well-posedness in L^2(R) for the full scaling-subcritical range.

Constructed the propagator S_V for rough time-dependent potentials.

Mass conservation implies global solutions for real-valued symbols.

Abstract

In this paper, we study a class of one-dimensional nonlocal nonlinear Schr\"odinger 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 Schr\"odinger equation and the Calogero--Moser derivative nonlinear Schrd\"oinger 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.