Maria Colombo - Instability and non-uniqueness for the Euler and Navier-Stokes equations 1 ; Maria Colombo - Instability and non-uniqueness for the Euler and Navier-Stokes equations 1: Summer School 2023 - New Trends in mathematical fluid dynamics

We start introducing some basic facts about the Navier-Stokes equations, such as weak solutions, Leray global solutions, forward and backward self-similar solutions. We then explore recent developments in understanding the fundamental question of whether Leray-Hopf solutions to Navier-Stokes equations are unique. Following Jia-Sverak and Guillod-Sverak program, we describe how non-uniqueness can follow from instability in self-similarity variables. We then discuss a recent work in collaboration with Albritton and Brue’, where two distinct Leray solutions with zero initial velocity and identical body force are built. This nonuniqueness result builds in turn on Vishik's answer to another long-standing uniqueness problem about the 2D Euler equations.