Mathematics Colloquia and Seminars

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The dynamics of a passive scalar, such as temperature or concentration, transported by an incompressible flow can be modeled by the advection-diffusion equation. Advection often results in the formation of complicated, small-scale structures and can result in solutions dissipating energy at a rate much faster than the corresponding heat equation in regimes of weak diffusion. This phenomenon is typically referred to as enhanced dissipation. If the velocity field is sufficiently rough, e.g., only Holder continuous with some exponent strictly less than one (as is expected in regimes of turbulent advection), then it is possible to have anomalous dissipation. That is, for the rate of energy dissipation to become independent of the small diffusivity parameter. In this talk, I will first discuss a recent joint work with Tarek Elgindi and Jonathan Mattingly in which we construct an example of optimal enhanced dissipation on the two-dimensional torus using time-periodic, alternating piece-wise linear shear flows. I will then briefly describe a joint work with Tarek Elgindi where we prove anomalous dissipation for a related alternating shear construction.