On anomalous diffusion

Vlad Vicol, New York University
Fine Hall 214

Anomalous diffusion is the fundamental ansatz of phenomenological theories of passive scalar turbulence (e.g. Richardson, Taylor, Obukhov, Corrsin, Batchelor, Kraichnan). As with the anomalous dissipation of kinetic energy in a turbulent fluid, the anomalous dissipation of passive scalar energy in a turbulent flow, as the Reynolds and Peclet numbers diverge, has been confirmed numerically and experimentally to an extraordinary extent. A satisfactory theoretical explanation of this phenomenon is to date not available. In fact, the mathematical analysis seems to lag the phenomenological theories by so much, that not even satisfactory examples of anomalous diffusion of passive scalars are available. In this talk, I will discuss a joint work with Scott Armstrong in which we construct a class of incompressible vector fields that have many of the properties observed in a fully turbulent velocity field, and for which the associated scalar advection-diffusion equation generically displays anomalous diffusion. We also propose an analytical framework in which to study anomalous diffusion, via a backward cascade of renormalized eddy viscosities. Our proof is by "fractal" homogenization, that is, we perform a cascade of homogenizations across arbitrarily many length scales.