Global Solution and Blow-up of the Stochastic Nonlinear Schr\"{o}dinger system

TL;DR

This paper investigates the stochastic nonlinear Schrödinger system with multiplicative noise, establishing local and global well-posedness, conditions for global existence, and criteria for blow-up in energy space.

Contribution

It provides new results on well-posedness and blow-up phenomena for the stochastic NLS system using Strichartz estimates, including sharp blow-up criteria.

Findings

Global well-posedness in mass subcritical and defocusing cases.

Existence of solutions when initial $L^2$ norm is small.

Sharp criteria for blow-up phenomena.

Abstract

We study the stochastic Nonlinear Schr\"{o}dinger system with multiplicative white noise in energy space $H^1$. Based on deterministic and stochastic Strichartz estimates, we prove the local well-posedness and uniqueness of mild solution. Then we prove the global well-posedness in the mass subcritical case and the defocusing case. For the mass subcritical case, we also investigate the global existence when the $L^2$ norm of initial value is small enough. In addition, we study the blow-up phenomenon and give a sharp criteria.