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

TL;DR

This work addresses local well-posedness for the nonlinear Schrödinger equation with modulated dispersion on the torus, $i \partial_t u = \Delta u \frac{d w}{dt} + |u|^{2k}u$, under rough time modulation $w$. It adopts the deterministic Young integral framework and introduces $(\rho,\gamma)$-irregularity to capture irregular dispersion, combining this with divisor-counting multilinear estimates to control the nonlinear term. The authors establish a key $2k+1$-linear estimate, proving $X \in C^{\gamma}([0,T];\mathcal L_{2k+1}(H^s(\mathbb{T}^d)))$ for $s>\frac{d}{2}-\frac{\rho}{k}$, and apply a fixed-point argument to obtain a unique local solution $u \in C^{\lambda}([0,T];H^s(\mathbb{T}^d))$ with continuous dependence on initial data. This extends Chouk-Gubinelli’s results to higher dimensions and nonlinearities, providing a deterministic framework for rough-dispersion regimes and linking to regularization-by-noise phenomena.

Abstract

We consider the Cauchy problem of the nonlinear Schrödinger 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.

Theorems & Definitions (20)

• Definition 1
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.1
• Theorem 1.2
• Remark 1.4
• Remark 1.5
• Theorem 2.1: Young integral
• Theorem 2.2: Young solutions