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

Abstract

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.

Introduction to SPDEs from Probability and PDE

“…This programme was originally carried out by Leray, see [Ler34], and now constitutes an important result in the wider programme to establish existence and uniqueness of strong solutions to the Navier-Stokes equations. A review of Leray's work in English can be found in [Far17] and the preprint [OP17]. A fuller survey of results concerning the Navier-Stokes equations can be found in [Ser14a] (available as [Ser14b]).…”

Navier-Stokes blow-up rates in certain Besov spaces whose regularity exceeds the critical value by $ε\in (1,2]$

“…Leray [7] provides an outline of how to prove (1.1), using what is now known as the mild formulation of the Navier-Stokes equations. More recent proofs of (1.1) using the mild formulation include [2,4,5,9], where [9] provides full details of Leray's outline, while [2] extends (1.1) from the Lebesgue setting L p = L p,p to the general Lorentz setting L p,q . An altogether different proof of (1.1)p<∞ when n = 3 (under the assumption lim tրT u(t) L p (R 3 ) = ∞) is provided by Robinson and Sadowski [11]; their proof uses energy estimates, and is closely aligned with the spirit of this present paper.…”

“…We adopt the convention that F f (ξ) = R n e −iξ•x f (x) dx for f ∈ S. We recall that the Fourier transform of a compactly supported distribution is a smooth function 9. For example, if F u is locally integrable near ξ = 0, then u ∈ S ′ h .…”

Weak Solutions to the Navier–Stokes Inequality with Arbitrary Energy Profiles

“…The fundamental mathematical theory of the Navier-Stokes equations goes back to the pioneering work of Leray (1934) (see Ożański & Pooley (2017) for a comprehensive review of this paper in more modern language), who used a Picard iteration scheme to prove existence and uniqueness of local-in-time strong solutions. Moreover, Leray (1934) and Hopf (1951) proved a global-in-time existence (without uniqueness) of weak solutions satisfying the energy inequality, u(t) 2 + 2ν t s ∇u(τ ) 2 dτ ≤ u(s) 2 (1.2) for almost every s ≥ 0 and every t > s (often called Leray-Hopf weak solutions) in the case of the whole space R 3 (Leray) as well as in the case of a bounded, smooth domain Ω ⊂ R 3 (Hopf).…”

“…The fundamental mathematical theory of the Navier-Stokes equations goes back to the pioneering work of Leray (1934) (see Ożański & Pooley (2017) for a comprehensive review of this paper in more modern language), who used a Picard iteration scheme to prove existence and uniqueness of local-intime strong solutions. Moreover, Leray (1934) and Hopf (1951) proved the global-in-time existence (without uniqueness) of weak solutions satisfying the energy inequality,…”