Analysis of a Stratified Kraichnan Flow

We consider the stochastic convection-diffusion equation $${\mathrm{\partial }}_{t}u(t,\mathbf{x})=\nu \mathrm{\Delta }u(t,\mathbf{x})+V(t,x_1){\mathrm{\partial }}_{x_2}u(t,\mathbf{x}),$$ for $t>0$ and $\mathbf{x}=(x_1,x_2)\in {\mathbb{R}}^2$, subject to ${\theta }_0$ being a nice initial profile. Here, the velocity field $V$ is assumed to be centered Gaussian with covariance structure $$\text{Cov}[V(t,a),V(s,b)]={\delta }_0(t-s)\rho (a-b)\text{for all }s,t\ge 0\text{ and }a,b\in \mathbb{R},$$ 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 $\nu >0$. Our a priori estimates imply the physically-natural fact that, quite generally, the solution dissipates. In fact, very often, \begin{equation} P\left{\sup_{|x_1|\leq m}\sup_{x_2\in$\mathbb{R}$} |u(t,,{\bf x})| = O\left(\frac{1}{\sqrt t}\right)\qquad$\text{as }t\to \mathrm{\infty }$ \right}=1\qquad$\text{for all }m>0$, \end{equation} and the $O(1\mathrm{/}\sqrt{t})$ 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 $t\to \mathrm{\infty }$ and as $\nu\to 0$. 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.