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

Understand

This article offers a modern perspective which exposes the many contributions of Leray in his celebrated work on the Navier--Stokes equations from 1934.

• Although the importance of his work is widely acknowledged, the precise contents of his paper are perhaps less well known.
• The purpose of this article is to fill this gap.
• We follow Leray's results in detail: we prove local existence of strong solutions starting from divergence-free initial data that is 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.