Analysis Seminar

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Ergodicity, Positive Lyapunov Exponents, and Partial Damping for Random Switching

Speaker: Jonathan Mattingly, Duke University

Location: Warren Weaver Hall 1302

Date: Thursday, October 3, 2024, 11 a.m.

Synopsis:

I will consider some new models inspired by the PDEs/SPDE with complex dynamics such as the 2D Euler and Navier-Stokes equations. The models introduce randomness onto the system through a random splitting scheme. I will explain how the randomly split Galerkin approximations of the 2D Euler equations and other related dynamics can be shown to possess a unique invariant measure that is absolutely continuous with respect to the natural Liouville measure, despite the existence of other invariant measures corresponding to fix points of the PDEs. I will then explain how on proves that the dynamics with respect to this measure have positive Lyapunov exponents almost surely.

Lastly, I will discuss recent results that show that the system has a unique invariant measure even when damping is applied to part of the system.

All of these results require proving a version of the “Hörmander's Sum of square” theorem concerning hypoelliptic operators but rather for the composition of random maps.