Lagrangian chaos and scalar mixing in stochastic fluid mechanics (Sam Punshon-Smith -- Brown)

4:00 pm on Thursday, March 21, 2019
5:00 pm on Thursday, March 21, 2019
111 Cummington Mall - MCS 148
Lagrangian chaos refers to the chaotic behavior of Lagrangian trajectories in a fluid. This chaotic behavior often characterized by a positive Lyapunov exponent, namely the property that initially close trajectories will separate at an exponential rate after long time. In this talk we will consider a variety of stochastically forced fluid models, including the 2 dimensional Navier-Stokes equations, and show that under certain non-degeneracy conditions on the noise, the stochastic flow possesses a positive Lyapunov exponent. The proof crucially uses the theory of random dynamical systems as well as tools from Malliavin calculus and control theory to satisfy a certain non-degeneracy criterion originally due to Furstenberg. We will explore several important consequences of a positive exponent for passive scalars advected by the fluid. For instance, as a corollary it is straight forward to rigorously prove Yaglom's law, an analogue of Kolmogorov's famous 4/5 law for passive scalar turbulence. More importantly, one can show almost sure exponential decay in time of passive scalars in any negative Sobolev norm (exponential mixing) by using the positive exponent to show geometric ergodicity of the two point Lagrangian motion.