Next Seminar

The seminar will resume on September 29th, Thursday, 14:00-16:00 UTC.

Sam Punshon-Smith (Tulane University):

Using regularity to estimate Lyapunov exponents


I will discuss a new method for estimating Lyapunov exponents for hypoelliptic diffusions from below using quantitative local regularity estimates on the stationary measure of the lift to the projective bundle. For damped driven truncations of nonlinear conservative SPDE in fluctuation dissipation scaling, I will outline a general strategy for proving positivity of the top Lyapunov exponent that can be applied to Galerkin truncations of the stochastic Navier-Stokes equations on T^2. The proof in the Navier-Stokes case involves verifying transitivity properties of a certain matrix Lie algebra associated with the linearized Euler equations using tools from algebraic geometry. The techniques are quite general and can be applied more broadly to Galerkin truncations of many other evolution type SPDE. This work is joint with Jacob Bedrossian and Alex Blumenthal.

Martina Hofmanova (Bielefeld University):

Non-unique ergodicity for deterministic and stochastic 3D Navier--Stokes and Euler equations


We establish existence of infinitely many stationary solutions as well as ergodic stationary solutions to the three dimensional Navier--Stokes and Euler equations in the deterministic as well as stochastic setting, driven by an additive noise. The solutions belong to the regularity class $C(\mathbb{R};H^\vartheta)\cap C^\vartheta(\mathbb{R};L^2)$ for some $\vartheta>0$ and satisfy the equations in an analytically weak sense. Moreover, we are able to make conclusions regarding the vanishing viscosity limit and anomalous dissipation. The result is based on a new stochastic version of the convex integration method which provides uniform moment bounds locally in the aforementioned function spaces. Joint work with Rongchan and Xiangchan Zhu.

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