Improved quasi-invariance result for the periodic Benjamin-Ono-BBM equation

TL;DR

This paper extends the quasi-invariance results of Gaussian measures for the periodic Benjamin-Ono-BBM equation to its full well-posedness range, overcoming critical dispersion challenges with advanced analytical techniques.

Contribution

It provides the first comprehensive quasi-invariance result for the full well-posedness range of the BO-BBM equation, combining recent methodological advances.

Findings

Extended quasi-invariance to full well-posedness range

Developed new long-time integrability bounds

Overcame critical dispersion challenges

Abstract

We extend recent results of Genovese-Luca-Tzvetkov (2022) regarding the quasi-invariance of Gaussian measures under the flow of the periodic Benjamin-Ono-BBM (BO-BBM) equation to the full range where BO-BBM is globally well-posed. The main difficulty is due to the critical nature of the dispersion which we overcome by combining the approach of Coe-Tolomeo (2024) with an iteration argument due to Forlano-Tolomeo (2024) to obtain long-time higher integrability bounds on the transported density.