Nonlinear Schr\"odinger equations with spatial white noise potential on full space for $d\le 3$

TL;DR

This paper establishes existence, uniqueness, and dispersive properties of solutions to nonlinear Schrödinger equations with spatial white noise in up to three dimensions, using advanced calculus techniques.

Contribution

It provides the first results on propagation without loss of regularity and localization for such equations in full space, including the three-dimensional case.

Findings

Proved existence and uniqueness of energy solutions for $d\,\leq 3$.

Established Strichartz inequalities for dispersive properties.

Achieved local well-posedness for low regularity solutions.

Abstract

In this paper, we prove existence and uniqueness of energy solutions for nonlinear Schr\"odinger equations with a multiplicative white noise on $R^d$ with $d\le3$. We rely on an exponential trans-form and conserved quantities for existence of energy solutions. Using paracontrolled calculus, we prove Strichartz inequalities which encode the dispersive properties of the solutions. This allows to obtain local well-posedness for low regularity solutions and uniqueness of energy solutions for various equations. In particular, our results are the first results of propagation without loss of both regularity and localization for such equations in full space as well as the first results on $R^3$ for such singular dispersive SPDEs. We are also obtain local well-posedness in two dimensions for deterministic initial data.