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

TL;DR

This work addresses local wellposedness for a 2D Gross-Pitaevskii equation with rough spatial noise by constructing a renormalized operator $-A$ via an enhanced noise $\Xi=(Y,Z)$ and a gauge transform $\rho=e^Y$. It develops a rigorous functional-analytic framework based on Hermite-Sobolev spaces, weighted Besov spaces, and paradifferential calculus, enabling a paracontrolled treatment of the singular terms in the renormalized operator. A key contribution is the derivation of Strichartz estimates for the renormalized propagator, which drive a contraction argument proving local wellposedness for nonlinearities of the form $|u|^{2\gamma}u$ with $2\gamma<1/(2\kappa)$. The paper also establishes conservation laws (mass and energy) and global existence in defocusing cases or for small data in focusing regimes, advancing the understanding of Anderson-type nonlinear Schrödinger equations in unbounded domains and highlighting a noise-independent propagation of regularity in the initial data.

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 Schrödinger 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.

Theorems & Definitions (70)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Remark 2.2
• Proposition 2.3
• Lemma 2.4
• Corollary 2.5
• Proposition 2.6
• Corollary 2.7
• Proposition 2.8