Venue: MCM210
Title: Nonlinear PDEs with modulated dispersion-regularization by noise
Abstract: In this talk, we consider the Korteweg-de Vries equation (KdV) and related equations such as the Benjamin-Ono equation (BO) with a modulated dispersion. In particular, we demonstrate regularization-by-noise phenomena in the following three ways.
(i) We establish well-posedness of the modulated KdV on both the circle and real line in the regime where the unmodulated KdV is ill-posed. In particular, we show that the modulated KdV on the circle is locally well-posed in Sobolev spaces of arbitrarily low regularity, provided that the modulation is sufficiently irregular.
(ii) While BO exhibits quasilinear behavior, we show that sufficiently irregular modulations "semilinearize" the equation by proving its local well-posedness via a contraction argument.
(iii) We establish nonlinear smoothing for these modulated equations, where a gain of regularity of the nonlinear part can be (arbitrarily) larger for more irregular modulations.
If time permits, I will discuss more recent results on the stochastic modulated KdV on the circle with multiplicative noise, where we exhibit a new type of regularization-by-noise phenomenon on the stochastic term. This talk is based on joint works with Khalil Chouk (formerly Edinburgh), Massimiliano Gubinelli (Oxford), Tadahiro Oh (Edinburgh), Jiawei Li (Edinburgh), and Andreia Chapouto (Versailles).
