Diffusion-Approximation for the Advection-Diffusion of a Passive Scalar by a Space-Time Gaussian Velocity Field

Abstract

We study the asymptotic behavior, as ∈ goes to zero, of a passive scalar T ∈ (x,t)solution of the following advection-diffusion equation:

$$\begin{array}{*{20}{c}} {\frac{{\partial {{T}^{ \in }}}}{{\partial t}} = \frac{v}{2}\Delta {{T}^{ \in }} + \frac{1}{ \in }V\left( {x,\frac{t}{{{{ \in }^{2}}}}} \right)\cdot \nabla {{T}^{ \in }},\;t > 0,} \\ {{{T}^{ \in }}\left( {x,0} \right) = {{T}_{0}}\left( x \right),\;x \in {{\mathbb{R}}^{d}},} \\ \end{array}$$

where v is a strictly positive diffusion constant and{V(x,t) : x ∈ ℝ^d, t ≥ 0 }is a mean zero homogeneous Gaussian field. We assume that the covariance is of the form

$$\mathbb{E}\left\{ {V\left( {x,t} \right)V*\left( {y,s} \right)} \right\} = \Gamma \left( {x - y} \right)\exp \left( { - a\left| {t - s} \right|} \right),$$

and under some mild regularity assumption on Γ, we prove that T ^∈ (x,t) converges in distribution to the solution of a stochastic partial differential equation. We derive the effective diffusion coefficient from this result. This work is a generalization of previous works by Bouc-Pardoux [3] and Kushner-Huang [8] where the velocity field is of the form $\frac{1}{ \in }V\left( {x,{{Z}_{{t/{{ \in }^{2}}}}}} \right)$ for some finite-dimensional ergodic noise process Z. Our situation is an example of infinite-dimensional noise.

This work is partially supported by a joint NSF-CNRS grant

Partially supported by ONR N00014-91-1010