Analysis of the Navier-Stokes regularity problem

众所周知，Navier-Stokes方程的“正则性”问题是我们这个时代最困难、最有趣、最重要的数学问题之一。它是克莱数学研究所在2000年提出的七个著名的千禧年奖问题之一，其中只有一个（庞加莱猜想）已经解决，并且一直是英国和国外许多国际会议和科学机构（如ERC（欧盟）和NSF（美国））授予的赠款的焦点。然而，相关问题的纳维-斯托克斯方程已经吸收了数学家的注意力，因为1934年时，关键的重大进展，迄今为止是由法国数学家让勒雷。早在1822年就推导出的非线性偏微分方程（PDE）的Navier-Stokes系统被认为给出了近似确定流体（如空气和水）运动的规则。这其中有很多问题，不仅是方程是否准确地描述了基础物理，而且可能更根本的是方程本身是否可以给出物理相关的预测。他们可能会预测潜在的偏心和纯粹的数学运动，包含被称为数学“奇点”的不稳定行为。当然，根据我们对自然界的经验，我们认为如果没有显著的外力作用，这种情况在现实中永远不会发生，因此知道奇点是否能在数学上形成，将对模型的有效性产生根本性的影响。勒雷证明了方程总是提供至少一个预测的运动，数学家们一直试图确定这些方程是否真的包含数学奇点。此外，这些都是非线性偏微分方程分析中的典型问题（往往基于物理学、生物学、金融学和社会学等领域的现象），对这个问题的解决方案也将对开发强大的分析工具产生重大的数学影响。事实上，EPSRC最近在英国牛津和爱丁堡建立了两个大型的分析和偏微分方程研究中心，以及两个博士培训中心，以提高英国在这些数学领域的竞争力。此外，在英国，特别是在牛津，沃里克，伦敦和苏塞克斯，已经做了很多工作来解决和提高对Navier-Stokes和其他分析非线性偏微分方程问题的兴趣。这个项目的具体目标是简化最近解决Navier-Stokes正则性问题的大量努力，并确定有效的未来方向，以及把问题本身和数学奇点的想法，变成一个更令人满意的环境这些目标可以大致分为三个部分：（1）表征潜在的奇异性，（2）理解现有技术的障碍，（3）在推导方程时使用的建模假设的背景下探索“奇异性”。这些努力将对潜在奇点的性质给出更全面和定性的描述。然后，人们可以利用这种理解来集中精力，要么构造奇点的明确例子，要么排除它们形成的可能性，以及将数学问题本身置于令人满意的建模环境中。

The question of "regularity" for the Navier-Stokes equations is well-known to be one of the most difficult, interesting and important mathematical questions of our time. It is the subject of one of the seven famous Millennium Prize Problems posed by the Clay Mathematics Institute in 2000, of which only one (the Poincare Conjecture) has since been resolved, and has been as well the focus of numerous international conferences and grants awarded in the UK and abroad by scientific agencies such as the ERC (EU) and NSF (USA). However the relevant questions for the Navier-Stokes equations have absorbed the attention of mathematicians since 1934 when the key significant advance to date was made by French mathematician Jean Leray. The Navier-Stokes system of nonlinear partial differential equations (PDEs), derived as early as 1822, is thought to give rules that approximately determine the motion of fluids such as air and water. There are many issues with this, not only as to whether the equations accurately describe the underlying physics but perhaps more fundamentally whether the equations themselves could give physically relevant predictions. They may rather predict potentially eccentric and purely mathematical motions containing erratic behaviors known as mathematical "singularities". Of course, from our experience with nature we expect that without significant external forcing this would never happen in reality, and so knowing whether or not singularities could form mathematically would have fundamental implications for the efficacy of the model. Leray proved that the equations always provide at least one predicted motion, and mathematicians have been trying to determine whether these could in fact contain mathematical singularities ever since. These are moreover quite typical questions in the analysis of nonlinear PDEs (which tend to be based on phenomena occurring in areas such as physics, biology, finance and sociology) and a resolution to this question would also have a significant mathematical impact with regard to development of robust analytical tools. The EPSRC has in fact recently developed two large funded research centers for analysis and PDEs in the UK, in Oxford and Edinburgh, as well as two doctoral training centers, to try to boost the UK's competitiveness in these mathematical areas. Moreover, much work has been done in the UK, most notably in Oxford, Warwick, London and Sussex, to address and boost interest in the Navier-Stokes and other analytical nonlinear PDE problems.This project aims specifically to streamline the copious recent efforts to resolve the Navier-Stokes regularity problem and determine effective future directions, as well as to put the question itself, and the idea of mathematical singularities, into a more satisfying context. These goals can be grouped roughly into three parts: (1) characterizing potential singularities, (2) understanding obstacles to current techniques and (3) exploring "singularity" in the context of the modeling assumptions used in deriving the equations. These efforts will give a more comprehensive and qualitative description of the nature of potential singularities. One can then use this understanding to focus efforts to either construct explicit examples of singularities or rule out the possibility of their formation, as well as to put the mathematical question itself into a satisfying modeling context.