题目：Nonlinear PDEs with modulated dispersion – regularization by noise

报告人：李国鹏  北京理工大学    

摘要：We study dispersive equations with a time non-homogeneous modulation acting on the linear dispersion term. In this talk, we consider the Korteweg-de Vries equation (KdV) and related equations such as the Benjamin-Ono equation (BO) and the intermediate long wave equation (ILW). By imposing irregularity conditions on the modulation, we demonstrate phenomena known as regularization by noise in the following three ways:

(i) For sufficiently irregular modulation, we establish local well-posedness of the modulated KdV on both the circle and real line in settings where the unmodulated KdV is ill-posed. In particular, we show that the modulated KdV on the circle with a sufficiently irregular modulation is locally well-posed in  Sobolev spaces of arbitrarily low regularity. By combining the -method (from dispersive PDEs) and the sewing lemma (controlled rough paths), we also prove global well-posedness in negative Sobolev spaces.    

(ii) While equations like BO and ILW exhibit quasilinear behavior, we show that sufficiently irregular modulations semilinearize these equations by proving their local well-posedness via a contraction argument.

(iii) Finally, we show nonlinear smoothing for these modulated equations, where we show that a gain of regularity of the nonlinear part becomes (arbitrarily) larger for more irregular modulations.

This talk is based on joint work with Khalil Chouk (formerly UoE), Massimiliano Gubinelli (Oxford), Tadahiro Oh (UoE), and Jiawei Li (UoE).    

报告时间：2025年5月14日（周三）下午16:00-17:00

报告地点：教四楼112    

联系人：高传伟