On anomalous diffusion
ABSTRACT: 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.
BIO: Vlad Vicol received his Ph.D. from the University of Southern California in 2010. He held positions at the University of Chicago (postdoc) and Princeton University (assistant professor), and is currently a professor of mathematics at the Courant Institute of Mathematical Sciences, NYU. He was an Alfred P. Sloan Research Fellow, he is a joint recipient of the 2019 Clay Research Award, an invited speaker at the 2022 International Congress of Mathematicians, and a Fellow of the AMS, class of 2023. His research focuses on the analysis of nonlinear PDEs arising in fluid dynamics.