Dissipation Enhancement, Mixing and Blow-up Suppression

Diffusion and mixing are two fundamental phenomena that arise in a wide variety of applications. In this talk we quantitatively study the interaction between diffusion and mixing in the context of problems arising in fluid dynamics. The first question we address is how fast the energy can decay in the advection diffusion equation. Even though this is a simple linear equation, the energy decay rate is intrinsically related to the mixing properties of the advecting velocity field, and there are many unresolved open questions. I will present a few recent results involving both upper and lower bounds, and then consider applications to studying the long time dynamics of a few model non-linear equations.