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This article offers a modern perspective which exposes the many contributionsof Leray in his celebrated work on the Navier--Stokes equations from 1934.Although the importance of his work is widely acknowledged, the precisecontents of his paper are perhaps less well known. The purpose of this articleis to fill this gap. We follow Leray's results in detail: we prove localexistence of strong solutions starting from divergence-free initial data thatis either smooth, or belongs to $H^1$, $L^2\cap L^p$ (with $p\in(3,\infty]$),as well as lower bounds on the norms $\| \nabla u (t) \|_2$, $\| u(t) \|_p$($p\in(3,\infty]$) as $t$ approaches a putative blow-up time. We show globalexistence of a weak solution and weak-strong uniqueness. We present Leray'scharacterisation of the set of singular times for the weak solution, from whichwe deduce that its upper box-counting dimension is at most $\tfrac{1}{2}$.Throughout the text we provide additional details and clarifications for themodern reader and we expand on all ideas left implicit in the original work,some of which we have not found in the literature. We use some modernmathematical tools to bypass some technical details in Leray's work, and thusexpose the elegance of his approach.

Leray's fundamental work on the Navier-Stokes equations: a modern review of "Sur le mouvement d'un liquide visqueux emplissant l'espace"