Anomalous Diffusion by Fractal Homogenization

Abstract

For every \(\alpha < \nicefrac 13\), we construct an explicit divergence-free vector field \({\textbf {b}}(t,x)\) which is periodic in space and time and belongs to \(C^0_t C^{\alpha }_x \cap C^{\alpha }_t C^0_x\) such that the corresponding scalar advection-diffusion equation

$$\begin{aligned} \partial _t \theta ^\kappa + {\textbf {b}}\cdot \nabla \theta ^\kappa - \kappa \Delta \theta ^\kappa = 0\end{aligned}$$

exhibits anomalous dissipation of scalar variance for arbitrary \(H^1\) initial data:

$$\begin{aligned}\limsup _{\kappa \rightarrow 0} \int _0^{1} \int _{\mathbb {T}^d} \kappa \bigl | \nabla \theta ^\kappa (t,x) \bigr |^2 \,dx\,dt >0.\end{aligned}$$

The vector field is deterministic and has a fractal structure, with periodic shear flows alternating in time between different directions serving as the base fractal. These shear flows are repeatedly inserted at infinitely many scales in suitable Lagrangian coordinates. Using an argument based on ideas from quantitative homogenization, the corresponding advection-diffusion equation with small \(\kappa \) is progressively renormalized, one scale at a time, starting from the (very small) length scale determined by the molecular diffusivity up to the macroscopic (unit) scale. At each renormalization step, the effective diffusivity is enhanced by the influence of advection on that scale. By iterating this procedure across many scales, the effective diffusivity on the macroscopic scale is shown to be of order one.