Approximation of the Global Attractors of Evolution Equations

进化方程全局吸引子的近似

• 批准号: 9706903
• 负责人: Michael Jolly
• 金额: $ 17.41万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706903&HistoricalAwards=false
• 关键词: Approximation; Global; Attractors; Evolution; Equations

9706903 M. Jolly Abstract This project concerns the long time behavior of certain dissipative physical systems. One major component seeks to locate global attractors by interpolatory means rather than by direct numerical solution of initial value problems. It would extend our previous work which used a Taylor expansion in complexified time at a single point in phase space. The new approach will use several points, typically those on solutions provided either by independent computations or from experimental data. The main mathematical tool in this effort will be Nevalinna-Pick interpolation. The systems on which this will be tested include the 2-D Navier-Stokes (NS), Kuramoto-Sivashinsky (KS) Lorenz equations as well as simple geostrophic models. Another major component is to compute invariant manifolds to arbitrary accuracy. We will use the visualization of 2-D (un)stable manifolds to help understand the geometric mechanisms behind certain global bifurcations in 3-D phase space. This involves computing a major portion of global manifolds. Other applications require only that the manifold be computed along particular trajectories. In one case a center manifold will be treated in this way to compute bounded solutions to an elliptic partial differential equation (PDE) in an infinite cylinder. In another, the dimension of phase space for the KS equation will be effectively reduced to three dimensions by restricting the flow to an inertial manifold of that dimension. Such a reduction will allow us to study global bifurcations as described above. The algorithms developed to compute these manifolds will also be applied to the sets (conjectured to be manifolds) of a prescribed exponential growth rate backward in time for the NS equation. In fact we will construct such sets as stable "manifolds" for an inverted form of the NS equation in which infinity and the origin of phase space are swapped. These sets play a role in the interpolatory approach to locating global attractors, and thus bring our research full circle. The main purpose of this work is to develop reliable methods to determine whether certain dynamic behavior in physical systems is permanent, or merely temporary. The ultimate application will be to climatology. Since the earth's weather system has been evolving for millions of years, one would expect that unless sudden external events take place, the patterns we are living through now will more or less continue for a reasonable period of time. This is not about accurate long time forecasting, rather it is about confirming basic assumptions regarding the mathematical models used in making those predictions. The scientific community makes a tremendous effort in deriving appropriate mathematical equations, and discretizing them so they can be solved on a computer, all to produce a function of time, which should describe some aspect of the weather. We all know how often this computed function of time deviates from the actual weather after a relatively short time period. The major source of this error is not clear. Is it in the model itself? Is it from the numerical approximation in the computer solution? Or is it that both the model and the approximation are valid, but the actual solution is very sensitive to small changes in the initial data, and we simply need to tighten the tolerance of error in that data and in the algorithm used at each time step. Our work is directed at distinguishing between the first two cases and the third. Indeed we seek to validate the model-algorithm pair which produces the forecast, as producing a pattern which is of a permanent nature, even if it is not the particular pattern we are experiencing after several days time. The failure of such a test will indicate that either the model and/or the method of solution are faulty. This approach can be applied to other physical problems. Indeed the initial testing of the methods will be done on systems less invol ved than that of the weather, but which are nevertheless of current scientific interest. In particular we consider fundamental models of combustion, fluid flow, and turbulence.

9706903 M. Jolly 摘要 该项目涉及某些耗散物理系统的长期行为。 其中一个主要组成部分寻求通过插值方法而不是通过初始值问题的直接数值求解来定位全局吸引子。 它将扩展我们之前的工作，即在相空间中的单个点上使用复杂时间的泰勒展开。 新方法将使用几个点，通常是通过独立计算或实验数据提供的解决方案。 这项工作的主要数学工具是 Nevalinna-Pick 插值法。 将对其进行测试的系统包括二维纳维-斯托克斯 (NS)、Kuramoto-Sivashinsky (KS) 洛伦兹方程以及简单的地转模型。 另一个主要组成部分是计算任意精度的不变流形。 我们将使用 2-D（不稳定）稳定流形的可视化来帮助理解 3-D 相空间中某些全局分岔背后的几何机制。 这涉及计算全局流形的主要部分。 其他应用仅需要沿着特定轨迹计算流形。 在一种情况下，将以这种方式处理中心流形，以计算无限圆柱体中椭圆偏微分方程 (PDE) 的有界解。 另一方面，通过限制流向该维度的惯性流形，KS 方程的相空间维度将有效地减少到三维。 这种减少将使我们能够研究如上所述的全局分叉。 为计算这些流形而开发的算法也将应用于 NS 方程的向后指定指数增长率的集合（推测为流形）。 事实上，我们将把这样的集合构造为 NS 方程的倒置形式的稳定“流形”，其中无穷大和相空间的原点被交换。 这些集合在定位全局吸引子的插值方法中发挥着作用，从而使我们的研究回到了原点。 这项工作的主要目的是开发可靠的方法来确定物理系统中的某些动态行为是永久的还是暂时的。 最终的应用将是气候学。 由于地球的天气系统已经进化了数百万年，人们预计，除非发生突然的外部事件，否则我们现在所经历的模式或多或少将持续一段合理的时间。这不是关于准确的长期预测，而是关于确认用于进行这些预测的数学模型的基本假设。 科学界付出了巨大的努力来推导适当的数学方程，并将它们离散化，以便可以在计算机上求解，所有这些都是为了产生时间函数，该函数应该描述天气的某些方面。 我们都知道，在相对较短的时间段后，计算出的时间函数与实际天气的偏差有多频繁。 该错误的主要来源尚不清楚。 它在模型本身中吗？ 是由计算机解法中的数值近似得出的吗？ 或者模型和近似值都是有效的，但实际的解决方案对初始数据的微小变化非常敏感，我们只需要收紧该数据和每个时间步使用的算法的误差容忍度。 我们的工作旨在区分前两种情况和第三种情况。 事实上，我们试图验证产生预测的模型-算法对，因为它产生了一种具有永久性的模式，即使它不是我们几天后经历的特定模式。 这种测试的失败将表明模型和/或解决方法有问题。 这种方法可以应用于其他物理问题。 事实上，这些方法的初步测试将在与天气相比较少涉及的系统上进行，但这仍然具有当前的科学兴趣。 我们特别考虑燃烧、流体流动和湍流的基本模型。