Invariant Gibbs Measures and a.s. Global Well-Posedness for Coupled KdV Systems

TL;DR

This paper proves the invariance of Gibbs measures and almost sure global well-posedness for a family of coupled KdV systems on the torus, under certain conditions on the coupling parameter and Diophantine constraints.

Contribution

It establishes the invariance of Gibbs measures and global well-posedness for coupled KdV systems for a range of coupling parameters, extending previous results.

Findings

Gibbs measure invariance under coupled KdV flow for specified parameters

Almost sure global well-posedness in the periodic setting

Conditions on the coupling parameter and Diophantine properties are crucial

Abstract

We continue our study of the well-posedness theory of a one-parameter family of coupled KdV-type systems in the periodic setting. When the value of a coupling parameter \alpha \in (0, 4) \setminus 1, we show that the Gibbs measure is invariant under the flow and the system is globally well-posed almost surely on the statistical ensemble, provided that certain Diophantine conditions are satisfied.