Stochastic Perturbations in the Fractional Nonlinear Schr\"odinger Equation: Well-posedness and Blow-up

TL;DR

This paper studies the effects of stochastic perturbations on the fractional nonlinear Schrödinger equation, establishing well-posedness, global existence, and blow-up criteria, highlighting how noise influences solution behavior.

Contribution

It introduces stochastic Strichartz estimates for fractional Schrödinger equations and demonstrates how multiplicative noise affects blow-up and global solutions.

Findings

Local well-posedness in energy-subcritical regime

Global existence via stochastic mass and energy evolution

Noise suppresses blow-up in supercritical cases

Abstract

This work investigates radial solutions for nonlinear fractional Schr\"odinger equations driven by multiplicative noise. Leveraging radial deterministic and stochastic Strichartz estimates, we establish local well-posedness in the energy-subcritical regime for the stochastic fractional nonlinear Schr\"odinger equation. Global existence is subsequently demonstrated through stochastic evolution of mass and energy. In focusing supercritical settings, we derive blow-up criteria via localized virial inequality, revealing how multiplicative noise measurably suppresses blow-up formation compared to deterministic dynamics.