**Title:** Optimal mixing by incompressible flows

**Speaker:** Anna Mazzucato, Pennsylvania State University

**Abstract: ** I will discuss transport of passive scalars by incompressible flows, such as a die is mixed in a fluid or milk is stirred in coffee. This is an important process in many physical systems, from transport of pollutants in the atmosphere to to cell processes in the human body. This process can be mathematically modeled by solutions of linear partial differential equations, called transport equations.

Constructing efficient mixers has practical applications in industry, e.g. in extrusion, but it is an interesting problem in mathematics as well, with connections to several other areas, for instance dynamical systems, geometric measure theory, and optimal control. A flow will be called an optimal mixer is it achieves mixing in the optimal amount of time. I will introduce quantitative measures of mixing and stirring under physical constraint on the flow. In particular, I will present recent results concerning examples that achieve the optimal theoretical rate of mixing in the case of flows with a prescribed bound on derivatives of the associated velocity, such as under an energy or a power budget. As a byproduct, our analysis showcases interesting geometry and topology allowed in so-called regular Lagrangian flows. There are flows with velocity of Sobolev, but not Lipschitz, regularity. Such pathologies also lead to examples of (instantaneous) loss of regularity for solutions to linear transport equations with velocity in this class.