Speaker: Franco Flandoli

Abstract: Standing the fact that many fluid mechanic equations do not have uniqueness of weak solutions, or regular solutions are not known to be global, in the last two decades there has been an intense activity to understand whether random perturbations may regularize the equations.

In finite dimensions, or for linear PDEs with a Lagrangian side, several results of restoration of uniqueness by noise have been proved; they will be illustrated in the first part of the lectures. However, for nonlinear PDEs they do not work well and several counterexamples have been proved.

On the contrary, it seems that noise may help to prevent blow-up for certain equations, possibly including the vorticity field of 3D Navier-Stokes equations. These delays of blow-up results will be described in the second part of the lectures.

Speaker: Dallas Albritton

Abstract: 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., Brué, 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.