[Paper Review] 2D-Defocusing Nonlinear Schrödinger Equation with Random Data on Irrational Tori

Chenjie Fan, Yumeng Ou+2 more|arXiv (Cornell University)|Oct 8, 2019

Advanced Mathematical Physics Problems7 references4 citations

TL;DR

This paper extends Bourgain's seminal result on the invariance of the Gibbs measure for the 2D defocusing cubic nonlinear Schrödinger equation from rational to arbitrary irrational tori. By developing new counting lemmata for lattice point sets in quadratic forms with irrational parameters and adapting probabilistic and Strichartz-type estimates, the authors establish almost sure local well-posedness and global flow invariance of the Gibbs measure on general 2D tori, resolving a long-standing open problem in stochastic nonlinear dispersive equations.

ABSTRACT

We revisit the work of Bourgain on the invariance of the Gibbs measure for the cubic, defocusing nonlinear Schrödinger equation in 2D on a square torus, and we prove the equivalent result on any tori.

Motivation & Objective

• To extend Bourgain's result on Gibbs measure invariance for the 2D defocusing cubic NLS from rational to general (irrational) tori.
• To overcome the breakdown of number-theoretic counting techniques used in rational tori by developing new lattice point estimates valid for irrational quadratic forms.
• To reprove the almost sure local and global well-posedness of the NLS with random initial data in the support of the Gibbs measure on arbitrary tori.
• To demonstrate that weaker counting bounds—still sufficient—can replace the sharp number-theoretic bounds used in the original rational torus proof.
• To provide a complete, detailed reworking of Bourgain's argument with explicit remarks on the challenges arising from irrationality.

Proposed method

• Developed new counting lemmata for the number of lattice points in regions of the form $\{ (x,y) \in \mathbb{Z}^2 : x^2 + \alpha y^2 \leq R^2 + O(1) \}$ with irrational $\alpha > 0$, replacing the sharp number-theoretic bounds used in rational tori.
• Adapted Bourgain's probabilistic framework, including Wick ordering and large deviation estimates, to the general torus setting.
• Employed refined Strichartz-type estimates in the $X^{s,b}$-space framework, with time localization and frequency decomposition.
• Used a Cauchy-Schwarz-type inequality (Lemma C.1) to control stochastic averages over random Fourier coefficients.
• Applied the $L^3$-based estimate (Lemma 3.7) and Hölder/Cauchy-Schwarz arguments in time and frequency variables to control nonlinear interactions.
• Reconstructed the full proof of local almost sure well-posedness with explicit attention to the loss of periodicity in time on irrational tori.

Experimental results

Research questions

• RQ1 Can the invariance of the Gibbs measure for the 2D defocusing cubic NLS be established on irrational tori, where the time-periodicity of solutions fails?
• RQ2 What alternative counting techniques can replace the sharp number-theoretic bounds on lattice points in $x^2 + a y^2 = R^2$ when $a$ is irrational?
• RQ3 Is the probabilistic method of local well-posedness via random data still viable when the deterministic Strichartz estimates are weaker than in the rational case?
• RQ4 Can the original Bourgain argument be fully adapted to general tori without relying on the periodicity of linear solutions?
• RQ5 Does the absence of rational frequency ratios on irrational tori still allow for sufficient control of nonlinear interactions in the Gibbs measure framework?

Key findings

• The authors establish the invariance of the Gibbs measure for the 2D defocusing cubic NLS on any 2D torus, including irrational ones, extending Bourgain's original result on the square torus.
• New counting lemmata are developed that bound the number of lattice points in irrational quadratic forms with a power savings loss, sufficient for the probabilistic argument.
• The local almost sure well-posedness result holds for initial data in the support of the Gibbs measure, which lies in $H^s(\mathbb{T}^2)$ for $s < 0$, despite the solution flow being only known to be locally well-posed for $s > 0$.
• The proof is fully detailed and self-contained, with explicit remarks highlighting the modifications needed when rationality is absent.
• The authors confirm that the weaker counting bounds, though not as sharp as in the rational case, are still sufficient to close the probabilistic argument and establish measure invariance.
• The result confirms that the improved Strichartz estimates on irrational tori (as shown by Deng, Germain, and Guth) do not alter the necessity of probabilistic methods for reaching the low-regularity Gibbs measure support.