On Strichartz estimates for a dispersion modulated by a time-dependent deterministic noise

TL;DR

This paper investigates a nonlinear Schrödinger equation with dispersion modulated by a deterministic self-affine noise, deriving modified Strichartz estimates that lead to global well-posedness, illustrating regularization by noise in a deterministic setting.

Contribution

It introduces a novel approach to establish Strichartz estimates for equations with deterministic noise modulation, proving global well-posedness for supercritical nonlinearities.

Findings

Modified Strichartz estimates for noise-modulated dispersion

Global well-posedness for L2-supercritical nonlinearities

Demonstration of regularization by deterministic noise

Abstract

We address the Cauchy problem for a nonlinear Schr{\"o}dinger equation where the dispersion is modulated by a deterministic noise. The noise is understood as the derivative of a self-affine function of order H $\in$ (0, 1). Due to the self-similarity of the noise, we obtain modified Strichartz estimates which enables us to prove the global well-posedness of the equation for L2-supercritical nonlinearities. This is an occurence of regularization by noise in a purely deterministic context.