## EVENT DETAILS AND ABSTRACT

**Analysis Seminar**
**Title:** Lagrangian chaos, mixing, and scalar turbulence in stochastic fluid mechanics

**Speaker:** Sam Punshon-Smith

**Speaker Info:** Brown University

**Brief Description:**

**Special Note**:

**Abstract:**

The long-time behavior of a passive scalar in a fluid (like dye or some chemical concentration) has long been of interest in Physics, with things like mixing and the equilibrium distribution of power spectra of principle interest. In this talk I will discuss several recent rigorous results in this area for a passive scalar that is advected by the stochastic Navier-Stokes equations. We will see how tools from theory of random dynamical systems and the ergodic and hypoelliptic theory for stochastic PDE can be used to show that the associated Lagrangian flow has a positive Lyapunov exponent (Lagrangian chaos). An important non-trivial consequence of this is that any passive scalar almost surely mixes exponentially fast (chaotic mixing) and that the associatedÂ drift-diffusion equation almost surely dissipates its L^2 norm at an optimal ~ |log(kappa)| exponential time scale (enhanced dissipation), kappa being the strength of diffusion. Applications to passive scalar turbulence will be presented including a rigorous proof of Batchelor's -1 law on the power spectrum of passive scalars in equilibrium and the kappa to 0 limit of statistically stationary solutions to the forced drift-diffusion equation.

**Date:** Monday, November 04, 2019

**Time:** 4:10pm

**Where:** Lunt 105

**Contact Person:** Robin Neumayer

**Contact email:** neumayer@math.northwestern.edu

**Contact Phone:** 847-491-5580

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