This material has been published in Recent Progress in the Theory of the Euler and Navier-Stokes Equations edited by Robinson, James C. and Rodrigo, José L. and Sadowski, Witold, et al. This version is free to view and download for personal use only. Not for re-distribution, re-sale or use in derivative works. A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP url' above for details on accessing the published version and note that access may require a subscription.

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 ∩ L p (with p ∈ (3, ∞]), as well as lower bounds on the norms ∇u(t) 2 , u(t) p (p ∈ (3, ∞]) as t approaches a putative blow-up time. We show global existence of a weak solution and weak-strong uniqueness. We present Leray's characterisation of the set of singular times for the weak solution, from which we deduce that its upper box-counting dimension is at most 1 2 . Throughout the text we provide additional details and clarifications for the modern 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 modern mathematical tools to bypass some technical details in Leray's work, and thus expose the elegance of his approach.

In 2000 Constantin showed that the incompressible Euler equations can be written in an "Eulerian-Lagrangian" form which involves the back-to-labels map (the inverse of the trajectory map for each fixed time). In the same paper a local existence result is proved in certain Hölder spaces C 1,µ .We review the Eulerian-Lagrangian formulation of the equations and prove that given initial data in H s for n ≥ 2 and s > n 2 + 1, a unique local-in-time solution exists on the n-torus that is continuous into H s and C 1 into H s−1 . These solutions automatically have C 1 trajectories.The proof here is direct and does not appeal to results already known about the classical formulation. Moreover, these solutions are regular enough that the classical and Eulerian-Lagrangian formulations are equivalent, therefore what we present amounts to an alternative approach to some of the standard theory. * BCP is supported by an EPSRC Doctoral Training Award. † B.C.Pooley@warwick.ac.uk

How to cite:Please refer to published version for the most recent bibliographic citation information.