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

Yu Deng, Andrea R. Nahmod, Haitian Yue

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. 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.

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

Abstract

We consider the defocusing nonlinear Schrödinger equation on 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. 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.

Paper Structure

This paper contains 39 sections, 21 theorems, 341 equations.

Table of Contents

  1. Introduction
  2. Setup and the main theorem
  3. Wick ordering and Gibbs measure
  4. The main theorem
  5. A review of previous results
  6. Invariant measures
  7. Probabilistic well-posedness theory
  8. Difficulties and the strategy
  9. The probabilistic scaling
  10. Discussions of earlier methods
  11. Random averaging operators
  12. The full ansatz
  13. Further discussions
  14. Wave equations
  15. Other dispersion relations and/or nonlinearities
  16. ...and 24 more sections

Key Result

Proposition 1.1

Let $n$ be a nonnegative integer. Then almost surely in $u$ with respect to the Wiener measure $\mathrm{d}\rho$, the limit exists in $H^{0-}(\mathbb{T}^2)$. We will denote this limit by $W^n(u)$.

Theorems & Definitions (54)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Definition 1.10
  • ...and 44 more