Questions in Nonlinear Partial Differential Equations

该项目专注于描述流体流动的偏微分方程组的研究。这些方程在许多科学和工程领域发挥着重要作用，它们是今天用于模拟流体的计算机代码的基础，从天气预报到飞机和汽车设计，以及任何其他应用。从数学上讲，这些方程是出了名的难解，即使我们使用大型计算机，我们的计算机模型仍然不像我们希望的那样可靠。例如，将这种情况与计算复杂刚性结构中的应力进行比较是很有趣的。应力计算由一组不同的方程控制，在许多情况下，它们可以非常精确地完成。其中一个原因是刚性结构中的基本应力方程在数学上得到了更好的理解。流体计算的困难基本上是双重的。首先，解决方案本质上是复杂的，无论我们的理论有多好。其次，我们的理论从根本上是不完整的，因此我们必须在非常困难的计算环境中处理大量的不确定性。虽然我们对第一个困难无能为力，但第二个困难可以通过提高我们对方程的数学理解来解决。研究的最终目标是了解解决方案的行为，以使我们能够为解决方案设计更好的算法。好的理论在求解微分方程式中的作用，可以在一个更简单的行星系统运动方程的求解问题上得到很好的说明。在这种情况下，如果我们利用运动方程的更深层次的数学特性，我们可以显著扩展算法的精度，并对更长的时间尺度进行预测。有了流体方程，情况就更困难了，因为最终我们不能指望在模拟海流或大型水轮机中的快速流动时跟踪每一滴水。最后，我们不得不使用一些平均化，而正确选择平均化是最困难的部分。在更高的技术层面上，研究将集中在以下几个方面：二维流体的小尺度和长时间行为的产生，摄动理论的极限，模型方程，以及二维湍流和部分阻尼。作为例子，我们将描述我们在湍流研究方面的计划。这项工作将在二维环境中(在数学层面上)检验我们最好的湍流理论。(3D问题目前还遥不可及)。目前，湍流理论是基于一些关于平均化的启发式假设。试探法能从数学上证明是合理的吗？在更技术的层面上，如果我们去掉几个高傅里叶模式上的粘性阻尼，会发生什么？启发式湍流理论预测，这几乎不会对流体的整体行为产生任何影响(在湍流区域)。由于最近数学方法的进步，这个问题现在可能是触手可及的(尽管它仍然很困难)。剩下的问题同样错综复杂，但首席调查员已经为每个问题开发了新的方法，预计将取得重大进展。

The project focuses on the studies of partial differential equations describing fluid flows. These equations play an important role in many areas of science and engineering, and they are at the basis of computer codes used today for modeling fluids, from weather prediction to aircraft and car design, and any number of other applications. Mathematically, the equations are notoriously difficult to solve and even if we use large computers, our computer models are still not as reliable as we would like. It is interesting to compare the situation for example with calculating the stress in complicated rigid structures. The stress calculations are governed by a different set of equations, and in many cases, they can be done with very good precision. One of the reasons is that the underlying equations for stress in rigid structures are much better understood mathematically. The difficulties with fluid calculations are essentially two-fold. First, the solutions are intrinsically complicated, regardless of how good our theory is. Second, our theory is fundamentally incomplete, and therefore we have to deal with a lot of uncertainty in a very difficult computational environment. While there is not much we can do about the first difficulty, the second difficulty can be addressed by improving our mathematical understanding of the equations. The ultimate goal of the research is to understand the behavior of the solutions to the degree that we would be able to design better algorithms for the solutions. The role of good theory in solving differential equations can be well illustrated on a simpler problem of solving equations of motion for planetary systems. In that case, if we use deeper mathematical properties of the equations of motion, we can significantly extend the precision of the algorithms, and make predictions for much longer time-scales. With fluid equations, the situation is more difficult, because ultimately we cannot hope to follow every drop of water when simulating, say, an ocean current, or a fast flow in a large water turbine. In the end, we have to use some averaging, and the right choice of averaging is the most difficult part. A good mathematical knowledge of the theoretical issues surrounding the fluid equations is important for achieving these goals.At a more technical level, the research will focus on the following areas: the generation of small scales and long-time behavior of 2d fluids, the limits of perturbation theory, model equations, and 2-d turbulence and partial damping. By way of example we will describe our program in the study of turbulence. This work will test (at the mathematical level) our best theories of turbulence, in the 2d environment. (The 3d problems are currently out of reach). Currently, the turbulence theory is based on some heuristic assumptions about averaging. Can the heuristics be justified mathematically? At a more technical level, what happens if we remove viscous damping on a few high Fourier modes? The heuristic turbulence theory predicts that this will have hardly any effect on the overall behavior of the fluid (in a turbulent regime). Due to recent advances in mathematical methods, this problem may now be within reach (although it is still difficult). The remaining problems are similarly intricate, however the principal investigator has developed new methods for each of them and it is expected that significant progress will be made.