Gianluca Crippa (University of Basel) will offer an exclusive lecture series to the SPP2410 on
Introduction to the theory of mixing for incompressible flows
It will take place in the week 16.-19.06.2025 at 10:15-12:00 and will be made available as video meeting.
Introduction to the theory of mixing for incompressible flows
Mixing in fluid flows is a ubiquitous phenomenon, which arises in many situations ranging from physical processes, to industrial processes, to everyday occurrences (such as mixing of cream in coffee). In my lectures, I will provide a gentle introduction to the theory of mixing from a PDE point of view, in which the main question is to provide universal bounds on the decay of a suitable notion of mixing scale for a passive scalar advected by an incompressible field, and to understand the sharpness of such bounds. I will address the following topics:
- The continuity equation and the flow of a vector field.
- Mixing and mixing scales (geometric and analytical).
- Lower bounds on the mixing scales for Lipschitz vector fields.
- A short introduction to the DiPerna-Lions theory.
- Energy estimates and (non optimal) lower bounds for the analytical mixing scale.
- Mild regularity of the regular Lagrangian flow.
- Exponential lower bound for the geometric mixing scale.
- Scaling analysis in self-similar evolutions and optimality of the exponential lower bounds.