Program

Self-similarity is a key ingredient in recent work on singularity formation in the Euler equations (Elgindi; Hou and Chen) and non-uniqueness in the Euler equations (Bressan, Murray, and Shen; Vishik) and Navier-Stokes equations (Jia, Sverak, and Guillod; A., Brue, and Colombo), with contributions by many more. The goal of this course is to familiarize the audience with the kind of self-similarity analysis which appears in the above works and will continue to play an important role in the development of the field.

Ideal incompressible fluid motion is a subject which is of both mathematical and physical significance. Mathematically, paths of fluid motion are geodesic in the (infinite dimensional) group of volume preserving diffeomorphisms. Physically, in an appropriate generalized sense, they describe the infinite Reynolds number limit of viscous fluids, capturing turbulent phenomena. We will discuss some foundational aspects of fluid turbulence, including guiding experimental observations, Kolmogorov’s 1941 theory on the structure of a turbulent flow, Onsager’s 1949 conjecture on anomalous dissipation and weak Euler solutions, as well as Landau’s Kazan remark concerning intermittency. Mathematical constraints on, as well as constructions that exhibit features of turbulent behavior will be discussed.

Lecture 1: Geometry of ideal fluid motion
Lecture 2: Long time behavior of 2D fluids
Lecture 3: Rigorous results on 3D turbulence

In the last years a new paradigm ("modern mathematics") has emerged. Here, the influence of computers has been crucial. In particular, computers have helped provide rigorous (computer-assisted) proofs in the context of partial differential equations, and machine learning has become a very useful tool. I will also describe new exciting future directions in the field.

Macroscopic models of collective behaviour—typically formulated as systems of conservation laws—describe the evolution of averaged density and velocity of interacting agents. While analogous to classical hydrodynamic equations derived from gas particle interactions, these models account for more complex inter-agent dynamics beyond simple collisions.

This lecture series will begin with a derivation of hydrodynamic models of collective behaviour from microscopic particle systems, highlighting the differences between these models and the classical Euler and Navier-Stokes equations of fluid dynamics. In the second and third lectures, we will focus on analytical techniques used to study the existence, uniqueness, and long-time behaviour of solutions in one spatial dimension. The final lecture will address the more challenging case of multi-dimensional systems, where weaker solution concepts—such as distributional or measure-valued solutions—are required.