The Navier-Stokes equations: functional analysis and dynamical systems

Navier-Stokes方程是流体流动的数学模型。但是，尽管它们被广泛用于流体流动各个方面的理论和计算分析，但它们的数学基础仍然不确定。2000年，克莱数学研究所宣布了一份七个千年问题的清单，每个问题的解决方案都将吸引100万美元的奖金。在这个列表中包括'经典问题'，如黎曼假设和庞加莱猜想（现在由佩雷尔曼的工作解决）;但在这里也可以找到存在的问题，（或其他）三维Navier-Stokes方程的唯一解。数学模型的要点是它能够预测：如果你知道初始时刻发生了什么，你就可以预测未来会发生什么。然而，能够做出“预测”依赖于模型只有一个解：从同一初始设置开始的两个（或更多）解使预测成为占卜而不是科学。（可以精确地给出正确的数学语言），对于三维Navier-Stokes方程仍然没有解决：虽然经常使用，但没有数学证据证明它们具有任何预测能力。这个建议的一部分集中在与这个基本困难有关的问题上，这是一条贯穿数学流体动力学的断层线。“奇点”的形成是预测能力丧失的过程，本项目将考虑如何限制这些奇点的形成（如果它们真的发生）。与此相关的问题是如何将Navier-Stokes方程与Euler方程联系起来，Euler方程是一个忽略粘性影响的更古老、更简单的模型。该提案的另一半考虑了考虑二维Navier-Stokes方程时出现的问题。二维模型的物理相关性较低，但不存在困扰三维模型的基本问题：这使其成为最终应用于三维情况的技术的有用测试平台。动力系统理论（混沌理论是其中的一部分）可以应用于二维方程。在这种情况下，可以证明方程有一个有限维的吸引子。以一种非常松散的方式，这意味着“从长远来看发生的事情应该相对容易描述”;在物理学语言中，人们可以将其表达为“充分发展的二维湍流具有有限的自由度”。对这个想法进行严格（且数学上具体）的解释构成了这个提议的另一半。

The Navier-Stokes equations are well established as the mathematical model for the flow of fluids. But while they are used extensively in both theoretical and computational analyses of every aspect of fluid flow, their mathematical foundations are still uncertain.In the year 2000, the Clay Mathematics Institute announced a list of Seven Millennium problems, solutions for each of which will attract a prize of one million dollars. Included in this list are 'classic problems' such as the Riemann Hypothesis and the Poincar conjecture (now solved by the work of Perelman); but here one can also find the question of the existence (or otherwise) of unique solutions for the three-dimensional Navier-Stokes equations.The point of a mathematical model is that it enables prediction: if you know what happens at an initial time, you can predict what will happen in the future. However, being able to make a 'prediction' relies on the model having only one solution: two (or more) solutions starting from the same initial setup make prediction a matter of divination rather than science.This is the 'uniqueness problem' (which can be formulated precisely given the correct mathematical language) that remains unresolved for the three-dimensional Navier-Stokes equations: although used routinely, there is no mathematical proof that they have any predictive power. Part of this proposal focuses on questions related to this fundamental difficulty, which is a fault line running through mathematical fluid dynamics. The formation of a 'singularity' is the process by which predictive power can be lost, and this project will consider how one can limit the formation of these singularities (should they actually occur). Related to this is the question of how the Navier-Stokes equations relate to the Euler equations, an older and some sense simpler model that neglects the effect of viscosity.The other half of the proposal considers questions that arise when one considers the two-dimensional Navier-Stokes equations. The two-dimensional model has less physical relevance, but does not suffer from the fundamental problems that bedevil its three-dimensional counterpart: this makes it a useful testbed for techniques that could eventually be applied in the three-dimensional case.The theory of dynamical systems (of which 'chaos theory' forms a part) can be applied to the two-dimensional equations. In this context, it is possible to show that the equations have an attractor that is finite-dimensional. In a very loose way this says that 'what happens in the long run should be relatively easy to describe'; in the language of physics one might express this as 'fully-developed two-dimensional turbulence has a finite number of degrees of freedom'.Giving a rigorous (and mathematically concrete) interpretation of this idea forms the other half of this proposal.