Continuity of the flow of KdV with regard to the Wasserstein metrics and application to an invariant measure

TL;DR

This paper proves the continuity of the KdV flow on probability measure spaces with respect to Wasserstein metrics, facilitating the study of invariant measures for the equation.

Contribution

It establishes the continuity of KdV flow in Wasserstein metrics on spaces of probability measures, linking dynamics and invariant measures.

Findings

Continuity of KdV flow in Wasserstein metrics on probability spaces.

Existence of invariant measures in the considered metric spaces.

Application to invariant measure analysis for KdV.

Abstract

In this paper, we prove the continuity of the flow of KdV on spaces of probability measures with respect to a combination of Wasserstein distances on $H^s$, $s>0$ and $L^2$. We are motivated by the existence of an invariant measure belonging to the spaces onto which these distances are defined.