We consider the stochastic convection-diffusion equation [ partial_t u(t,,{bf x}) = uDelta u(t,,{bf x}) + V(t,,x_1)partial_{x_2}u(t,,{bf x}), ] for $t>0$ and ${bf x}=(x_1,,x_2)inmathbb{R}^2$, subject to $theta_0$ being a nice initial profile. Here, the velocity field $V$ is assumed to be centered Gaussian with covariance structure [ text{Cov}[V(t,,a),,V(s,,b)]= delta_0(t-s)rho(a-b)qquadtext{for all $s,tge0$ and $a,binmathbb{R}$}, ] where $rho$ is a continuous and bounded positive-definite function on $mathbb{R}$. We prove a quite general existence/uniqueness/regularity theorem, together with a probabilistic representation of the solution that represents $u$ as an expectation functional of an exogenous infinite-dimensional Brownian motion. We use that probabilistic representation in order to study the It^o/Walsh solution, when it exists, and relate it to the Stratonovich solution which is shown to exist for all $ u>0$. Our a priori estimates imply the physically-natural fact that, quite generally, the solution dissipates. In fact, very often, begin{equation} Pleft{sup_{|x_1|leq m}sup_{x_2inmathbb{R}} |u(t,,{bf x})| = Oleft(frac{1}{sqrt t}right)qquadtext{as $ttoinfty$} right}=1qquadtext{for all $m>0$}, end{equation} and the $O(1/sqrt t)$ rate is shown to be unimproveable. Our probabilistic representation is malleable enough to allow us to analyze the solution in two physically-relevant regimes: As $ttoinfty$ and as $ uto 0$. Among other things, our analysis leads to a macroscopic multifractal analysis of the rate of decay in the above equation in terms of the reciprocal of the Prandtl (or Schmidt) number, valid in a number of simple though still physically-relevant cases.
Download