Time Evolution of a Passive Scalar in a Turbulent Incompressible Gaussian Velocity Field

Passive scalar equation is considered in a turbulent homogeneous incompressible Gaussian velocity field. The turbulent nature of the field results in non-smooth coefficients in the equation. A strong, in the stochastic sense, solution of the equation is constructed using the Wiener Chaos, and the properties of the solution are studied. The results apply to both viscous and conservative motions. 1. Passive Scalar in a Gaussian Field We consider the following transport equation to describe the evolution of a passive scalar θ in a random velocity field v: (1.1) θ̇(t, x) = 0.5ν∆θ(t, x)− v(t, x) · ∇θ(t, x) + f(t, x); x ∈ R, d > 1. Our interest in this equation is motivated by the on-going progress in the study of the turbulent transport problem (E and Vanden Eijnden [3], Gawȩdzki and Kupiainen [4], Gawȩdzki and Vergasola [5], Kraichnan [7], etc.) We assume in (1.1) that v = v(t, x) ∈ R, d ≥ 2, is an isotropic Gaussian vector field with zero mean and covariance E(v(t, x)v(s, y)) = δ(t− s)C(x− y) with some matrix-valued function C = (C(x), i, j = 1, . . . , d). It is well-known (see, for example, LeJan [8]) that in the physically interesting models the matrix-valued function C = C(x) has the Fourier transform Ĉ = Ĉ(z) given by Ĉ(z) = A0 (1 + |z|2)(d+α)/2 ( a zz |z|2 + b d− 1 ( I − zz T |z|2 )) , where z is the row vector (z1, . . . , zd), z is the corresponding column vector, |z|2 = zz, I is the identity matrix; α > 0, a ≥ 0, b ≥ 0, A0 > 0 are real numbers. Similar to [8], we assume that 0 < α < 2. By direct computation (cf. [1]), the vector field v = (v, . . . , v) can be written as (1.2) v(t, x) = ∑ k≥0 σ k(x)ẇk(t), 2000 Mathematics Subject Classification. Primary 60H15; Secondary 35R60, 60H40.