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

Abstract

We study the Gibbs dynamics for the Zakharov-Yukawa system on the two-dimensional torus T2, namely a Schrödinger-wave system with a Zakharov-type coupling (−Δ)γ. We first construct the Gibbs measure in the weakly nonlinear coupling case (0≤γ<1). Combined with the non-construction of the Gibbs measure in the strongly nonlinear coupling case (γ=1) by Oh, Tolomeo, and the author (2020), this exhibits a phase transition at γ=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≤γ<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.