A path of understanding fluid equations: from Leray to Ladyzhenskaya, and beyond

The mathematical theory of incompressible fluids, Ladyzhenskaya’s favorite topic, still poses challenges for us today. We briefly review the pioneering work of Leray on weak solutions for the Navier-Stokes equation (NSE), and the fundamental contributions of Ladyzhenskaya on the unique solvability of the NSE and the global attractor problem. We will then present some recent results on the regularity and large time behaviour of solutions for the NSE. Connections between these findings and Kolmogorov’s turbulence theory will be discussed as well.