The Navier-Stokes equations: functional analysis and dynamical systems

Navier-Stokes方程作为流体流动的数学模型已经被很好地建立起来。但是，尽管它们在流体流动的各个方面的理论和计算分析中得到了广泛的应用，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“千年七大难题”的名单，每个问题的答案都将获得100万美元的奖金。其中包括“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）；但这里也可以找到三维纳维-斯托克斯方程是否存在唯一解的问题。数学模型的意义在于它能够预测：如果你知道在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”依赖于只有一个解决方案的模型：从相同的初始设置开始的两个（或更多）解决方案使预测成为占卜问题，而不是科学问题。这就是三维纳维-斯托克斯方程仍未解决的“唯一性问题”（可以用正确的数学语言精确地表述）：尽管经常使用，但没有数学证据证明它们有任何预测能力。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（如果它们真的发生的话）。与此相关的问题是，纳维-斯托克斯方程如何与欧拉方程相关联，欧拉方程是一个更古老，更简单的模型，忽略了粘度的影响。提案的另一半考虑了在考虑二维Navier-Stokes方程时出现的问题。二维模型具有较少的物理相关性，但不会遭受困扰三维模型的基本问题：这使其成为最终可以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。以一种非常宽松的方式说，“长期发生的事情应该相对容易描述”；用物理学的语言，我们可以将其表达为“完全发展的二维湍流具有有限数量的自由度”。给这个想法一个严格的（和数学上具体的）解释形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.