Title: Stability and instability in incompressible fluids

Speaker: Michele Coti Zelati

Abstract: The study of stability of coherent structures in fluid flows dates back to the late 19th century and has fascinated mathematicians since Reynolds’ famous experiments in 1883. Mathematically, this problem can be phrased in terms of estimating the size of the basin of attraction of steady states of the Navier-Stokes equations and related models.

In Part I, I will review the theory of stability for shear flows, introducing the concepts of mixing, inviscid damping, and enhanced dissipation. We will see how the linear theory provides an explicit description of the asymptotic stability of certain steady states in the Euler and Navier-Stokes equations.

In Part II, I will discuss recent results on the stability of shear flows in the presence of additional stabilizing mechanisms: wave propagation. Such mechanisms appear in stratified or rotating fluids and can be quantified through the so-called transition threshold exponent, providing further insights into enhanced stability.