Two dimensional nonlinear Schr\"odinger equation with spatial white noise potential and fourth order nonlinearity

TL;DR

This paper establishes the global well-posedness and convergence of solutions for a two-dimensional nonlinear Schrödinger equation with spatial white noise and mixed nonlinearity, extending previous results to more complex nonlinearities.

Contribution

It extends prior work by proving global existence, uniqueness, and convergence for a 2D NLS with spatial white noise and nonlinearities between cubic and quartic, using regularization and renormalization techniques.

Findings

Proved global existence and uniqueness of solutions.

Established almost sure convergence of regularized solutions.

Extended previous results to more complex nonlinearities.

Abstract

We consider NLS on $\T^2$ with multiplicative spatial white noise and nonlinearity between cubic and quartic. We prove global existence, uniqueness and convergence almost surely of solutions to a family of properly regularized and renormalized approximating equations. In particular we extend a previous result by A. Debussche and H. Weber available in the cubic and sub-cubic setting.