Mathematics

The incompressible Navier-Stokes system is a fundamental mathematical model used to describe the motion of fluid flows. Despite being very old, our comprehension of this system remains limited.

The question of whether Navier-Stokes solutions develop singularities in finite time is still unresolved, making it one of the seven millennium prize problems.  

However, thanks to Leray's contributions, we are aware of the existence of weak solutions in the energy class that persist globally in time.

These solutions can be seen as potential extensions of classical solutions beyond the occurrence of a singularity.

Therefore, a crucial question arises: Are Leray-Hopf solutions unique? The aim of this course is to present recent advancements on this problem.

 

We will begin by providing an overview of the basics of Navier-Stokes, including dimensional analysis, energy balance, pressure, mild-solutions, and well-posedness results.

Subsequently, we will introduce the class of Leray solutions. We will establish a fundamental existence theorem, discuss partial regularity theorems, and prove a weak-strong uniqueness result.

Next, we will explore various aspects of the Jia, Sverak, and Guillod program, particularly focusing on how instability in self-similarity variables can lead to non-uniqueness of Leray solutions.

Lastly, we will present recent work (Albritton-B.-Colombo, Ann. Math. 2022) that rigorously establishes the non-uniqueness of Leray solutions with forcing. A central aspect of this work involves constructing a three-dimensional unstable vortex-ring. Along the way, we will discuss Vishik's results regarding the instability of two-dimensional vortices and applications to the study of the 2-dimensional Euler equation with vorticity in L^p.