Mixing in fluids: irregular transport, enhanced dissipation, and applications

Stirring and mixing in fluids, specifically incompressible fluids, have important consequences on many physical and biological processes, from dispersal of pollutants to transport of nutrients.  From a mathematical point of view, mixing can be studied in different contexts, from ergodic theory to homogenization.

In this talk, I will present a quantitative approach to mixing that arises in the analysis of partial differential equations. In this context, mixing is related to irregular transport by non-Lipschitz vector fields and, when combined with diffusion, it may lead to enhanced dissipation. A variety of techniques have been employed in the literature to study these mechanisms, from geometric analysis to optimal transport to spectral theory and probability.

I will first discuss examples of incompressible flows that mix optimally in time. Then, I will show how these examples lead to loss of regularity for solutions of transport equations. Lastly, if time allows, I will discuss enhanced dissipation and examples of flows that lead to enhanced dissipation for advection-(hyper)diffusion equations.