Work

Invariant Gibbs measures and global strong solutions for nonlinear Schrödinger equations in dimension two

Year: 2024

Type: article

Abstract: We consider the defocusing nonlinear Schrödinger equation on $\mathbb{T}^2$ with Wick ordered power nonlinearity, and prove almost sure global well-posedness with respect to the associated Gibbs measure. The heart of the matter is the uniqueness of the solution as limit of solutions to canonically truncated systems, and the invariance of the Gibbs measure under the global dynamics follows as a consequence. The proof relies on the novel idea of random averaging operators.

Source: Annals of Mathematics

DOI: 10.4007/annals.2024.200.2.1

Authors: Yu Deng, Andrea Nahmod, Haitian Yue

Institutions: University of Southern California, University of Massachusetts Amherst, ShanghaiTech University

Language: English

Topic: Advanced Mathematical Physics Problems

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: National Science Foundation

Awards: Probabilistic Aspects of Dispersive and Wave Equations, Propagation of Randomness in Nonlinear Evolution Equations, FRG: Collaborative Research: New Challenges in the Derivation and Dynamics of Quantum Systems