Super-diffusivity in a shear flow model from perpetual homogenization

Abstract

This paper is concerned with the asymptotic behavior solutions of stochastic differential equations $d{y}_t=d{\omega }_t-\nabla \Gamma (y_t)dt$, $y_0=0$ and $d=2$. $\Gamma$ is a $2\times 2$ skew-symmetric matrix associated to a shear flow characterized by an infinite number of spatial scales ${\Gamma }_{12}=-{\Gamma }_{21}=h(x_1)$, with $h(x_1)={\sum }_{n=0}^{\infty }{\gamma }_n{h}^n(x_1/R_n)$ where $h^n$ are smooth functions of period 1, $h^n(0)=0$, ${\gamma }_n$ and $R_n$ grow exponentially fast with $n$. We can show that $y_t$ has an anomalous fast behavior ($\E[|y_t|^2]\sim t^{1+ν}$ with $\nu >0$) and obtain quantitative estimates on the anomaly using and developing the tools of homogenization.

References in corpus (1)

• Anomalous Slow Diffusion from Perpetual Homogenization

Cited by in corpus (2)

• Averaging versus Chaos in Turbulent Transport?
• Passive tracer in a slowly decorrelating random flow with a large mean