Surface Stretching for Ornstein Uhlenbeck Velocity Fields

Abstract

The present note deals with large time properties of the Lagrangian trajectories of a turbulent flow in $R^2$ and $R^3$. We assume that the flow is driven by an incompressible time-dependent random velocity field with Gaussian statistics. We also assume that the field is homogeneous in space and stationary and Markovian in time. Such velocity fields can be viewed as (possibly infinite dimensional) Ornstein-Uhlenbeck processes. In d spatial dimensions we established the (strict) positivity of the sum of the largest $d-1$ Lyapunov exponents. As a consequences of this result, we prove the exponential stretching of surface areas (when $d=3$) and of curve lengths (when $d=2$.) This confirms conjectures found in the theory of turbulent flows.