Remark on the local well-posedness for NLS with the modulated dispersion

TL;DR

This paper establishes local well-posedness for the nonlinear Schrödinger equation with modulated dispersion across all spatial dimensions, using advanced integral and multilinear estimates, extending previous work by Chouk-Gubinelli.

Contribution

It generalizes the local well-posedness results for NLS with modulated dispersion to higher dimensions using Young integral theory and divisor counting techniques.

Findings

Proves local well-posedness for NLS with modulated dispersion in any dimension.

Extends the framework of Chouk-Gubinelli to broader settings.

Employs multilinear estimates based on divisor counting.

Abstract

We consider the Cauchy problem of the nonlinear Schr\"odinger equation with the modulated dispersion and power type nonlinearities in any spatial dimensions. We adapt the Young integral theory developed by Chouk-Gubinelli [K. Chouk and M, Gubinelli, Comm. Partial Differential Equations 40 (2015)] and multilinear estimates which are based on divisor counting, and show the local well-posedness. Our result generalizes the result by Chouk-Gubinelli.