Global well-posedness of the 2D nonlinear Schr\"odinger equation with multiplicative spatial white noise on the full space

TL;DR

This paper proves the global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space, extending previous results to arbitrary polynomial nonlinearities.

Contribution

It introduces a method to establish global solutions for the 2D NLS with multiplicative noise and arbitrary polynomial nonlinearity, building on gauge-transform techniques.

Findings

Established global well-posedness for the equation.

Extended previous results to full polynomial nonlinearities.

Constructed solutions as limits of approximating equations.

Abstract

We consider the nonlinear Schr\"odinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced by Hairer and Labb\'e (2015) and constructing the solution as a limit of solutions to a family of approximating equations. This paper extends a previous result by Debussche and Martin (2019) with a sub-quadratic nonlinearity.