Analysis Seminar

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Anomalous diffusion in the Kraichnan model and colored-in-time variants

Speaker: Keefer Rowan, New York University

Location: Warren Weaver Hall 1302

Date: Thursday, November 30, 2023, 11 a.m.

Synopsis:

We provide a complete, PDE-based proof of anomalous diffusion in the Kraichan model—a stochastic, white-in-time model of passive scalar turbulence. That is, we show an exponential rate of L²-diffusion, almost surely and in expectation, of a passive scalar advected by a certain white-in-time, correlated-in-space, divergence-free Gaussian field, with a rate uniform in the initial data and the diffusivity of the passive scalar. Additionally, we provide examples of correlated-in-time versions of the Kraichnan model which fail to exhibit anomalous diffusion despite their (formal) white-in-time limits exhibiting anomalous diffusion. As part of this analysis, we prove that anomalous diffusion of a scalar advected by some flow implies non-uniqueness of the ODE trajectories of that flow.