The Navier-Stokes equations: functional analysis and dynamical systems

流体流动的数学模型很好地建立在Navier-Stokes方程上。但是，尽管它们被广泛应用于流体流动的各个方面的理论和计算分析中，但它们的数学基础仍然不确定。2000年，克莱数学研究所公布了一份七个千年问题的清单，每个问题的解都将吸引100万美元的奖金。这份清单包括了一些经典的问题，如黎曼假说和庞卡尔猜想(现在由佩雷尔曼的工作解决了)；但在这里，人们也可以发现三维N-S方程存在(或不存在)唯一解的问题。数学模型的要点是它使预测成为可能：如果你知道初始时间发生了什么，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：从相同的初始设置开始的两个(或更多)解使得预测成为一种占卜而不是科学的问题。对于三维N-S方程来说，这是一个尚未解决的“唯一性问题”(给出正确的数学语言，它可以被精确地表达出来)：尽管经常使用，但没有数学证据表明它们具有预测能力。这项提案的一部分集中在与这一根本困难有关的问题上，这是贯穿数学流体动力学的一条断层线。奇点的形成是预测能力丧失的过程，这个项目将考虑如何限制这些奇点的形成(如果它们实际发生的话)。与此相关的是纳维斯托克斯方程如何与欧拉方程相联系的问题，欧拉方程是一个较旧的、在某种意义上更简单的模型，它忽略了粘性的影响。提案的另一半考虑了当人们考虑二维纳维斯托克斯方程时产生的问题。二维模型的物理相关性较小，但不存在困扰三维模型的基本问题：这使得它成为最终可能应用于三维模型的技术的有用试验台。动力系统理论(动力系统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。用一种非常宽松的方式来说，这就是“长期发生的事情应该相对容易描述”；在物理学的语言中，人们可以把这一点表述为“完全发展的二维湍流具有有限的自由度”。对这一概念进行严格(且在数学上具体)的解释形成了这一提议的另一半。

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.