Abstract

Partial differential equations (PDE) revolutionized science since the advent of calculus and serve as successful models to a staggeringly broad range of processes in nature and engineering.  Singularities in solutions of PDE are important since they often signify extreme phenomena or indicate a change in regime or breakdown of the model. I will overview some examples of singularity formation, with particular focus on fluid mechanics.  

Fluids are all around us, and we can witness the complexity and subtleness of their properties in every day life, in ubiquitous technology, and in dramatic events such as tornado or hurricane. There has been an enormous wealth of knowledge accumulated in the broad area of fluid mechanics, yet it is quite remarkable that many of the most fundamental and most important in applications questions remain poorly understood. In fact, Richard Feynman has called turbulence the greatest unsolved problem of classical physics, and one of the seven-million-dollar prize Millennium problems of Clay Institute is concerned with the possibility of singularities in the 3D Navier-Stokes equations – the foundational model of fluid mechanics. 

In this talk oriented towards non-experts, I will in particular discuss the connection between these two challenges, as well as other key themes in fluids and PDE.