At this colloquium (organized by Robin Chemnitz), we are happy to welcome:

Bill W. Cooperman (ETH Zürich)

Scalar mixing in random fluids

Pour a drop of milk into a cup of coffee. We know from experience that after a few moments of stirring, the mixture becomes homogeneous. In this talk, we study the advection of a passive scalar (the concentration of milk) under a random fluid (coffee) velocity on the torus. We prove, when the velocity solves the 2d stochastic Navier--Stokes equations on the torus, that the scalar converges exponentially to its mean. Our result applies even when only finitely many (as few as 4) Fourier modes are randomly forced. Joint with Keefer Rowan (EPFL).

Kathrin Padberg-Gehle (Universität Lüneburg)

From trajectories to transport: network-based approaches for coherent sets and scalar mixing in turbulent flows

Understanding, quantifying and controlling transport and mixing processes are central in the study of fluid flows. Many different Lagrangian approaches have been proposed for detecting organizing flow structures, including recent data-based methods that aim to identify such coherent objects directly from simulated or measured tracer trajectories. Among these approaches are Lagrangian flow networks, where trajectories serve as network nodes and the links are weighted according to spatio-temporal distances between trajectories. Spectral clustering as well as simple network measures such as node degrees or clustering coefficients can be used to identify flow regions of different dynamical behavior.  In this talk, we propose some extensions to the network-based framework that allow us (i) to study the long-term dynamics of coherent sets in turbulent flows and (ii) to describe and quantify the transport and mixing of scalar quantities. We demonstrate our approaches in a number of example systems, including turbulent Rayleigh-Bénard convection flows.

Sam Punshon-Smith (Tulane University)

The Spectrum of Turbulence: Lyapunov Exponents in Stochastic Fluid Dynamics

A central goal in the mathematical study of turbulence is to rigorously describe its chaotic nature using Lyapunov exponents. For the Stochastic Navier-Stokes (SNS) equations, this spectrum is fundamental: it is expected to control the system's predictability, its entropy, and the complexity of its global attractor. While this physical picture is compelling, it remains largely conjectural for SNS. This talk will survey this dynamical systems approach, focusing on recent rigorous progress where stochastic forcing is a key analytical tool. We will explore work on establishing chaos (a positive Lyapunov exponent), its connection to the fundamental limits of scalar mixing, and the underlying ergodic theory that provides a unified statistical picture for both. The talk is intended for a general mathematical audience and will introduce the foundational concepts needed to frame these results.