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Mathematical Physics Electronic Journal
ISSN 10866655
Volume 10, 2004
Paper 4
Received: Nov 4, 2003, Revised: Mar 3, 2004, Accepted: Mar 18, 2004
Editor: R. de la Llave
SPACE AVERAGES AND HOMOGENEOUS FLUID FLOWS
GEORGE ANDROULAKIS AND STAMATIS DOSTOGLOU
Abstract. The relation between space averages of vector fields in L1
loc(R3
) and averages
with respect to homogeneous measures on such vector fields is examined. The space
average, obtained by integration over balls in space, is shown to exist almost always and,
whenever the measure is ergodic or the correlation decays, to equal the ensemble average.
1. Introduction
In statistical theories of turbulence the velocity vector field u(t, x) of a fluid is taken for
each t > 0, x R3 to be a random variable on some (usually unspecified) probability space,
see [McC] for example. u(t, x) is also required to solve the NavierStokes equations in t and
x. In more mathematical formulations, the flow is described by a measure on a function
