School of Mathematical Sciences

Nonlinear PDEs with modulated dispersion – regularization by noise

Date(s)
Thursday 5th December 2024 (14:00-15:00)
Contact
Event Convenor Contact: William.Salkeld@nottingham.ac.uk
Description
Speaker's Name: Jiawei Li
Speaker's Affiliation: University of Edinburgh
Speaker's Research Theme(s): Statistics and Probability,Mathematical Physics
Abstract:
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 $I$-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), Guopeng Li (BIT) and Tadahiro Oh (UoE).

Venue: Chem X-2

School of Mathematical Sciences

The University of Nottingham
University Park
Nottingham, NG7 2RD

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