Invariant Gibbs dynamics for the two-dimensional Zakharov-Yukawa system

TL;DR

This paper constructs and analyzes the invariant Gibbs measure for the two-dimensional Zakharov-Yukawa system, establishing global well-posedness and measure invariance for weak nonlinear coupling, and identifying a phase transition at a critical coupling level.

Contribution

The paper introduces the construction of the Gibbs measure for the Zakharov-Yukawa system in the weakly nonlinear regime and proves almost sure global well-posedness and invariance, revealing a phase transition at the critical coupling.

Findings

Gibbs measure constructed for 0 ≤ γ < 1

Global well-posedness proven for 0 ≤ γ < 1/3

Invariance of Gibbs measure under dynamics established

Abstract

We study the Gibbs dynamics for the Zakharov-Yukawa system on the two-dimensional torus $\mathbb{T}^2$, namely a Schr\"odinger-wave system with a Zakharov-type coupling $(-\Delta)^\gamma$. We first construct the Gibbs measure in the weakly nonlinear coupling case ($0 \leq \gamma<1$). Combined with the non-construction of the Gibbs measure in the strongly nonlinear coupling case ($\gamma=1$) by Oh, Tolomeo, and the author (2020), this exhibits a phase transition at $\gamma = 1$. We also study the dynamical problem and prove almost sure global well-posedness of the Zakharov-Yukawa system and invariance of the Gibbs measure under the resulting dynamics for the range $ 0 \leq \gamma < \frac 13$. In this dynamical part, the main step is to prove local well-posedness. Our argument is based on the first order expansion and the operator norm approach via the random matrix/tensor estimate from a recent work Deng, Nahmod, and Yue (2020). In the appendix, we briefly discuss the Hilbert-Schmidt norm approach and compare it with the operator norm approach.