Asymptotics of Lyapunov Exponents and Phase Transitions for Fluids with Degenerate Forcing

Abstract

In this paper we analyze the Lyapunov exponents of a slow-fast system where the slow component is an Ornstein-Uhlenbeck process which perturbs the linear evolution of a fast variable through a bilinear form. These naturally arise in many finite-dimensional models for turbulence such as Galerkin truncation of 2D Navier-Stokes, the Lorenz 96 system, and the Lorenz 63 system. Using our general results about the slow-fast system, we are able to prove phase transitions in the ergodicity of each of these models when degenerate stochastic forcing is applied: as a parameter (e.g. noise strength or viscosity) varies, the number of invariant measures of the system switches from one to several. We are also able to obtain precise asymptotics for the top Lyapunov exponent associated to the unstable invariant measure. The crux of our proof is using a Wiener chaos expansion to show that mass quickly transfers from stable modes to unstable ones.