Title: Lower bounds on the Lyapunov exponent of linear PDEs driven by random velocity

Speaker: Jaeyun Yi (EPFL)

Abstract: Lyapunov exponents have been extensively studied in the context of fluid mechanics, particularly for the Euler and Navier-Stokes equations. In this regard, the lower bounds of Lyapunov exponents are known to serve as a benchmark for chaos and turbulence. We introduce two primary models in fluid mechanics: passive scalar advection and linearized stochastic Navier-Stokes equations, and prove the lower bounds of their Lyapunov exponents, which are quantitative in the diffusion parameter. Our result provides a typical filamentation length scale, which partially answers the Bachelor conjecture.  The proof relies on the high-frequency stochastic instability of the projective process via a detailed analysis of its spectral median. This is joint work with Martin Hairer, Sam Punshon-Smith, and Tommaso Rosati.