Systematic search for extreme and singular behaviour in some fundamental models of fluid mechanics

This review article offers a survey of the research program focused on a systematic computational search for extreme and potentially singular behaviour in hydrodynamic models motivated by open questions concerning the possibility of a finite-time blow-up in the solutions of the Navier–Stokes system. Inspired by the seminal work of Lu & Doering (2008 Ind. Univ. Math. 57, 2693–2727), we sought such extreme behaviour by solving PDE optimization problems with objective functionals chosen based on certain conditional regularity results and a priori estimates available for different models. No evidence for singularity formation was found in extreme Navier–Stokes flows constructed in this manner in three dimensions. We also discuss the results obtained for one-dimensional Burgers and two-dimensional Navier–Stokes systems, and while singularities are ruled out in these flows, the results presented provide interesting insights about sharpness of different energy-type estimates known for these systems. Connections to other bounding techniques are also briefly discussed. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.