Analysis of a Stratified Kraichnan Flow
Abstract
We consider the stochastic convectiondiffusion equation \[ \partial_t u(t\,,{\bf x}) =\nu\Delta u(t\,,{\bf x}) + V(t\,,x_1)\partial_{x_2}u(t\,,{\bf x}), \] for $t>0$ and ${\bf x}=(x_1\,,x_2)\in\mathbb{R}^2$, subject to $\theta_0$ being a nice initial profile. Here, the velocity field $V$ is assumed to be centered Gaussian with covariance structure \[ \text{Cov}[V(t\,,a)\,,V(s\,,b)]= \delta_0(ts)\rho(ab)\qquad\text{for all $s,t\ge0$ and $a,b\in\mathbb{R}$}, \] where $\rho$ is a continuous and bounded positivedefinite 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 infinitedimensional Brownian motion. We use that probabilistic representation in order to study the Itô/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 physicallynatural 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 $t\to\infty$} \right\}=1\qquad\text{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 physicallyrelevant regimes: As $t\to\infty$ 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 physicallyrelevant cases.
 Publication:

arXiv eprints
 Pub Date:
 November 2017
 arXiv:
 arXiv:1711.01650
 Bibcode:
 2017arXiv171101650H
 Keywords:

 Mathematics  Probability