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Invariant Gibbs measures and global Strong Solutions for periodic 2D nonlinear Schrödinger Equations.

Banff International Research Station for Mathematical Innovation and Discovery

In this talk we first give a quick background overview of Bourgain's approach to prove the invariance of the Gibbs measure for the periodic cubic NLS in 2D and of the para-controlled calculus of Gubinelli-Imkeller and Perkowski in the context of parabolic stochastic equations. We then present our resolution of the long-standing problem of proving almost sure global well-posedness (i.e. existence with uniqueness) for the periodic NLS in 2D on the support of the Gibbs measure, for any (defocusing and renormalized) odd power nonlinearity. Consequently we get the invariance of the Gibbs measure. This is achieved by a new method we call random averaging operators which precisely captures the intrinsic randomness structure of the problematic high-low frequency interactions at the heart of this problem. This is joint work with Yu Deng (USC) and Haitian Yue (USC).

Mathematics; Partial differential equations; Fluid mechanics

video/mp4

Author affiliation: University of Massachusetts

2020-11-02

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/76440

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/