Instability and non-uniqueness in fluid dynamics

Elia Bruè

This mini-course focuses on the instability and non-uniqueness of weak solutions to the incompressible Euler and Navier-Stokes equations in both two and three spatial dimensions. Two fundamental open problems in the field serve as focal points:

1. The uniqueness of Leray solutions to the three-dimensional Navier-Stokes equations.
2. The uniqueness and well-posedness of the two-dimensional Euler equations in vorticity formulation.

Recent advancements in addressing these challenges will be explored. The course is organized as follows: an initial lecture establishes foundational knowledge on weak solutions and existing well-posedness results. Subsequently, the focus shifts to Leray-Hopf solutions to the Navier-Stokes equations, covering self-similar solutions, instability, and non-uniqueness. In the third lecture, attention is directed towards the instability of two-dimensional vortices and Vishik's nonuniqueness theorem within the framework of the two-dimensional Euler equations with vorticity in L^p. The final lecture delves into the realms of flexibility and convex integration constructions within fluid dynamics.

Introduction to the theory of mixing for incompressible flows

Gianluca Crippa

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:

1. The continuity equation and the flow of a vector field.
2. Mixing and mixing scales (geometric and analytical).
3. Lower bounds on the mixing scales for Lipschitz vector fields.
4. A short introduction to the DiPerna-Lions theory.
5. Energy estimates and (non optimal) lower bounds for the analytical mixing scale.
6. Mild regularity of the regular Lagrangian flow.
7. Exponential lower bound for the geometric mixing scale.
8. Scaling analysis in self-similar evolutions and optimality of the exponential lower bounds.

Collective behavior: from particle to continuum models

Young-Pil Choi

Emergent aggregation and flocking phenomena that appear in many biological systems are simple instances of collective behavior. Recently, they have been extensively studied in various scientific disciplines such as applied mathematics, physics, biology, sociology, and control theory due to their biological and engineering applications. In my lectures, I will introduce several different types of microscopic models describing collective behaviors and discuss their applications. On the other hand, when the number of particles is very large, the microscopic description becomes computationally complicated. Thus, understanding how this complexity can be reduced is an important issue. Concerning this matter, I will address recent advances in the rigorous derivations from particles and the asymptotic limits connecting all the hierarchy of models in this active field of research, including kinetic models, pressureless Euler equations with nonlocal forces, and aggregation equations.

The well-posedness for the Navier-Stokes equations and its related equations in the maximal regularity class

Miho Murata

Abstract