Local wellposedness of the 2d Anderson-Gross-Pitaevskii equation

TL;DR

This paper establishes local wellposedness for a 2D Gross-Pitaevskii equation with rough potentials, including white noise, using a combination of operator construction, regularity analysis, and paracontrolled calculus.

Contribution

It introduces a novel approach to handle rough potentials in 2D Gross-Pitaevskii equations, including the construction of the Schrödinger operator and application of paracontrolled calculus.

Findings

Proved local wellposedness in 2D for equations with rough potentials.

Included potentials as rough as spatial white noise.

Developed a new framework combining operator theory and paracontrolled calculus.

Abstract

In this paper, the local wellposedness of a general Gross-Pitaevskii equation with rough potential is proven in dimension 2. The class of rough potentials we are considering is large enough to contain the spatial white noise and thus a renormalization procedure may be needed. We first construct the associated Schr\"odinger operator from its quadratic form. Then, the regularity of elements of its domain is explored. This allows to use a paracontrolled approach in order to obtain Strichartz estimates, which are used to prove the local wellposedness by a contraction argument.