Subtle statistical behavior in simple models for random advection-diffusion

J. Math. Sci. Univ. Tokyo
Vol. 1 (1994), No. 1, Page 23--70.

Horntrop, David J. ; Majda, Andrew J.
Subtle statistical behavior in simple models for random advection-diffusion
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Abstract:
Simple models for advection-diffusion with a statistical velocity field are studied here. These models involve advection by a time-independent random shear flow together with a constant mean flow. Several new and surprisingly subtle phenomena are developed here for the statistical behavior in these models. These new phenomena include: 1) mathematical criteria and examples with ill-posed evolution equations for the second order correlations and the mean statistics; 2) explicit sensitive dependence of the large scale, long time renormalization theory on parameters of the problem, such as the mean flow, the infrared cut-off, and the molecular diffusivity, for both the second order correlations and the mean statistics. This surprising sensitive dependence is explained in a self-consistent fashion both through mathematical theory and explicit examples.

Mathematics Subject Classification (1991): 39, 60
Mathematical Reviews Number: MR1298540

Received: 1993-09-02