题目: On measure transport for low-regularity nonlinear dispersive flows: A DiPerna–Lions viewpoint

报告人：孙晨旻 (LAMA,Université Paris-Est Créteil)

时间：7月22日（星期三）上午10点

地点：五教5306

摘要: The field of random-data problems for dispersive equations has its origins in the work of Lebowitz-Rose-Speer, Bourgain in the 1990s, concerning the invariant Gibbs measure program. Initiated by N. Tzvetkov in 2015, a generalization of this line of research focuses on studying the evolution of Gaussian measures under nonlinear dispersive dynamics, providing a natural candidate for out-of-equilibrium statistical mechanics for waves. A specific question is whether the transported Gaussian measures are quasi-invariant. If so, what can be said about the transported density? 

It turns out that this problem has a natural interpretation from the viewpoint of DiPerna–Lions transport theory, which is particularly useful in the rough, low-regularity setting. In particular, the transported density, if it exists, solves a Liouville equation in infinite-dimensional spaces. After explaining this abstract viewpoint, I will focus on a specific model, the 3D nonlinear wave equation, and explain the main ideas of the proof of a sharp quasi-invariance theorem for this model. This talk is based on joint work with Leonardo Tolomeo and Nikolay Tzvetkov.