Surface Stretching for Ornstein Uhlenbeck Velocity Fields
Rene Carmona, Princeton University
Stanislav Grishin, Princeton University
Lin Xu, Princeton University
Stanislav Molchanov, University of North Carolina at Charlotte
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.
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