Leray关于Navier-Stokes方程的基础性工作：对《论充满空间的黏性液体运动》的现代评述
Leray's fundamental work on the Navier-Stokes equations: a modern review of "Sur le mouvement d'un liquide visqueux emplissant l'espace"
AI总结：
本文系统梳理并补充了Leray 1934年关于Navier-Stokes方程的基础性工作，包括强解局部存在性、弱解全局存在性、弱-强唯一性及奇异时间集合的维数上界，以现代视角揭示其方法的完整性与优雅性。
摘要：
本文以现代视角展示Leray在其1934年关于Navier-Stokes方程的著名工作中所作出的诸多贡献。尽管其工作的重要性已获广泛认可，但其论文的确切内容或许并不广为人知。本文旨在填补这一空白。我们详细梳理Leray的结果：证明从光滑的、或属于$H^1$、$L^2\cap L^p$（其中$p\in(3,\infty]$）的无散度初始数据出发的强解的局部存在性，以及当$t$趋近假定的爆破时间时范数$\\| \nabla u (t) \\|_2$、$\\| u(t) \\|_p$（$p\in(3,\infty]$）的下界。我们证明弱解的全局存在性和弱-强唯一性。我们给出Leray对弱解奇异时间集合的刻画，并由此推出其上盒计数维数至多为$\tfrac{1}{2}$。全文为现代读者补充了额外的细节和澄清，并展开原工作中所有隐含的思想，其中一些在文献中尚未见到。我们使用一些现代数学工具绕过Leray工作中的某些技术细节，从而展现其方法的优雅之处。
英文摘要：
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. 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 $\tfrac{1}{2}$. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
