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

Speaker: Anna Mazzucato

Abstract: 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 dynamical systems 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

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, I will discuss enhanced dissipation and examples of flows that lead to enhanced
dissipation for advection-(hyper)diffusion equations, with applications to global existence of solutions to the two-dimensional Kuramoto-Sivashinsky equation, a model of long-
wave instability and flame front propagation.