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From Jean Leray to the millennium problem: the Navier–Stokes equations
Journal of Evolution Equations ( IF 1.4 ) Pub Date : 2020-11-09 , DOI: 10.1007/s00028-020-00645-3
Reinhard Farwig

One of the Millennium problems of Clay Mathematics Institute from 2000 concerns the regularity of solutions to the instationary Navier–Stokes equations in three dimensions. For an official problem description, we refer to an article by Charles Fefferman (Existence and smoothness of the Navier–Stokes equation. http://www.claymath.org/sites/default/files/navierstokes.pdf) on the whole space problem as well as the periodic setting in \(\mathbb {R}^3\). In this article, we will introduce the Navier–Stokes equations, describe their main mathematical problems, discuss several of the most important results, starting from 1934 with the seminal work by Jean Leray, and proceeding to very recent results on non-uniqueness and examples involving singularities. On this tour de force we will explain the open problem of regularity.



中文翻译:

从让·雷雷(Jean Leray)到千禧年问题:Navier–Stokes方程

从2000年开始,克莱数学学院的千年问题之一是关于三维固定Navier-Stokes方程解的规律性。有关正式问题的描述,请参阅Charles Fefferman的文章(Navier–Stokes方程的存在和平滑性。http://www.claymath.org/sites/default/files/navierstokes.pdf)以及\(\ mathbb {R} ^ 3 \)中的周期性设置。在本文中,我们将介绍Navier-Stokes方程,描述它们的主要数学问题,讨论几个最重要的结果,从1934年开始,让·雷雷(Jean Leray)进行了开创性的工作,然后发展到关于非唯一性和示例的最新结果涉及奇点。在这次巡回演出中 我们将解释规则性的开放性问题。

更新日期:2020-11-09
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