Colloquium
3:00 p.m., Friday (September 5, 2003)
Math Annex 1100
John Heywood
Mathematics, UBC
A curious phenomenon in a model problem,
suggestive of the hydrodynamic inertial range and smallest scale of motion
The Kolmogorov theory of turbulence, based on a scaling argument,
predicts a smallest significant length scale, beyond which a flow
is very smooth. Stated spectrally, there is a largest significant
wave number, beyond which the energy decays very rapidly with respect
to further increases in the wave number. However, this scaling argument
has never made contact with the rigorous mathematical theory of the
NavierStokes equations. If it were understood in the context of rigorous
theory it would have many important consequences, a particular
corollary being the global in time regularity of solutions of the
NavierStokes equations.
Here, we consider a certain infinite system of ordinary differential
equations, regarded as a highly simplified model of how energy might
be passed up the spectrum in the NavierStokes equations, into the
smaller scales of motion. Numerical experiments with this system of
equations reveal a very striking ``inertial range'' and ``smallest
scale'' phenomenon. One observes the apparent determination of a
``largest significant mode number'' by an abrupt change over just
a very few mode numbers in the character of the energy decay with
respect to mode number. We formulate corresponding mathematical
definitions and prove much of what is observed in these experiments,
especially regarding the energy decay of steady solutions with
respect to mode number. Our results for nonstationary solutions are
not as complete as for steady solutions, but their proofs are
probably more relevant to NavierStokes theory. We conclude by
describing the results of further experiments with related systems
of equations. In each of them the ``inertial range/smallest scale
phenomenon'' is found to persist. The results are quite dramatic,
and will be demonstrated during the talk with real time visualizations
of the computations. Finding that this phenomenon is generic to a
wide class of equations, we speculate on whether the system of
spectral NavierStokes equations belongs to it. Our objective is to
begin a rigorous investigation of smallest scale phenomena in simple
model problems, hoping for insights that might generalize to the
NavierStokes equations.
Refreshments will be served at 2:45 p.m. in the Faculty Lounge,
Math Annex (Room 1115).
