From Jean Leray to the millennium problem: the Navier-Stokes equations (Q2062843)

This survey article gives an overview of the history of research on the Navier-Stokes equations, from the earliest work until the most recent advances centered on the Millenium Problem. The author begins with the history and physical motivation of the equations and by discussing some of the work on the Navier-Stokes equations prior to \textit{J. Leray}'s groundbreaking work [Acta Math. 63, 193--248 (1934; JFM 60.0726.05)]. The author discusses Leray's work in some detail, as well as Serrin numbers, and the efforts at developing an appropriate formulation of strong solutions which was finished by \textit{T. Kato} [Math. Z. 187, 471--480 (1984; Zbl 0545.35073)]. The author continues by discussing recent advances in regularity criteria that involving the geometry of the flow, as well as the extension of regularity criteria into borderline and endpoint scale-invariant spaces. The author concludes by discussing Tao's recent blowup result for a model equation [\textit{T. Tao}, J. Am. Math. Soc. 29, No. 3, 601--674 (2016; Zbl 1342.35227)]. The survey article gives a thorough overview of the highly voluminous literature on the Navier-Stokes equation. It does an excellent job of discussing the methods involved -- the difficulties of each method, where they break down, etc. -- without either going into all the technical details or leaving out technical details entirely. This is a difficult balance to strike, so I would highly recommend this article to any graduate student working on fluid mechanics and looking to get the lay of the land on the Navier-Stokes regularity problem.