Lower Bounds on Lyapunov Exponents for Linear PDEs Driven by Stochastic Navier-Stokes

This lecture from the Analysis and Mathematical Physics series explores quantitative lower bounds for the top Lyapunov exponent of linear PDEs driven by two-dimensional stochastic Navier-Stokes equations on the torus. Discover recent work by Samuel Punshon-Smith (Tulane University) with Hairer, Rosati, and Yi that proves the top Lyapunov exponent for both advection-diffusion equations and linearized Navier-Stokes equations is bounded below by a negative power of the diffusion parameter κ, demonstrating that decay rates cannot be super-exponential. Learn about the first rigorous lower bound on the Batchelor scale in terms of diffusivity, which partially answers a conjecture by Doering and Miles. Understand the innovative concept of "high-frequency stochastic instability" that shows high-frequency states become unstable under stochastic perturbations, leading to a Lyapunov drift condition for the H1 norm over the L2 norm and quantitative estimates on decay rates. Explore the implications of these findings for mixing phenomena in fluid dynamics. The lecture will take place at 2:30pm in Simonyi Hall 101 with remote access available.