Barriers to Transport in Aperiodically Time-Dependent Two-Dimensional Velocity Fields: Nekhoroshev’s Theorem and “Almost Invariant” Tori

• Author: Stephen Wiggins, University of Bristol.


• Oral or poster: oral.


• Downloadable abstract: click here.


• Downloadable presentation/poster: click here.


• Abstract:
  Over the past 30 years the dynamical systems approach to transport in fluids has experienced continual growth, both in the scope of applications and in the development of computational and mathematical methods. Realistic velocity fields are not periodic in time (i.e. they are “aperiodic”) and are often defined as “data sets”, which results in them only being defined for a finite time. Both of these features--aperiodic time dependence and being only defined on a finite time interval--pose new computational and mathematical challenges to the dynamical systems approach to transport.
  
  In my talk I will discuss some of these aspects for two-dimensional, aperiodically time dependent flows. To begin with, I will given an overview of some of the main issues, and describe relevant work that does not appear known (and discuss some reasons for that). I will pay particular attention here to the “hyperbolic-elliptic” dichotomy in Hamiltonian systems.
  
  Kolmogorov-Arnold-Moser (KAM) tori are elliptic invariant manifolds that act as “complete” barriers to transport. While generalizing theorems on “hyperbolic objects and phenomena” is relatively straightforward (indeed, this was done many years before the dynamical systems approach to transport was developed, as I will describe) generalizing theorems related to “elliptic objects and phenomena”, such as KAM tori, has proven much more difficult. Nekhoroshev’s theorem is a finite time stability result that can be used to give meaning to the notion of an “almost invariant” torus. The finite time interval on which the estimates hold is exponentially long with respect to certain parameters of the system. In the words of Littlewood, “..while not eternity, this is a considerable slice of it”. I will explain the idea behind these exponential estimates, and I will also discuss the relation between the KAM and Nekhoroshev theorems and describe a Nekhoroshev theorem for two-dimensional, aperiodically time dependent velocity fields.
  
  If time permits, I will discuss some examples of two-dimensional, aperiodically time-dependent velocity fields where it can be shown analytically that they possess invariant tori. This allows one to see explicitly how invariant tori in aperiodically time-dependent velocity fields are manifested through standard diagnostics such as finite time Lyapunov exponents and the M function (this latter work is joint with Dr. Ana M. Mancho).