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抽象该项目涉及某些耗散物理系统的长时间行为。 一个主要组成部分试图通过插值手段来定位全球吸引子，而不是通过直接的初始价值问题的数值解决方案来定位全球吸引子。 它将扩展我们以前的工作，该工作在单个相空间的单个点上使用了泰勒膨胀的时间。 新方法将使用多个点，通常是由独立计算或实验数据提供的解决方案上的点。 这项工作中的主要数学工具将是Nevalinna-Pick插值。 将进行测试的系统包括2-D Navier-Stokes（NS），Kuramoto-Sivashinsky（KS）Lorenz方程以及简单的地质模型。 另一个主要组成部分是将不变的歧管计算为任意准确性。 我们将使用2-D（UN）稳定歧管的可视化来帮助了解3-D相空间中某些全局分叉背后的几何机制。 这涉及计算全球流形的主要部分。 其他应用仅要求沿特定轨迹计算流形。 在一种情况下，将以这种方式处理中心歧管，以计算无限缸中椭圆形偏微分方程（PDE）的有界溶液。 在另一个中，通过将流量限制为该维度的惯性歧管，KS方程的相空间维度将有效地降低到三个维度。 这样的减少将使我们能够如上所述研究全球分叉。 开发的算法是为了计算这些流形的算法，也将应用于NS方程的及时向后返回的规定指数增长率的集合（猜想为歧管）。 实际上，我们将为互换无限和相位空间的起源的NS方程的倒置形式构建稳定的“流形”集。 这些集合在定位全球吸引子的插值方法中发挥了作用，从而使我们的研究完整圈子。 这项工作的主要目的是开发可靠的方法，以确定物理系统中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发展了数百万年，因此人们希望，除非发生突然的外部事件，否则我们现在所生活的模式或多或少会持续一段合理的时间。这不是准确的长时间预测，而是关于确认有关用于做出这些预测的数学模型的基本假设。 科学界在得出适当的数学方程式并将其离散化，以便在计算机上解决这些方程式，以产生时间的函数，以描述天气的某些方面。 我们都知道，在相对较短的时间段之后，时间的计算功能偏离了实际天气。 此错误的主要来源尚不清楚。 是在模型本身中吗？ 是来自计算机解决方案中的数值近似值吗？ 或者是模型和近似值都是有效的，但是实际解决方案对初始数据的小变化非常敏感，我们只需要拧紧该数据和每个时间步中使用的算法的错误公差即可。 我们的工作旨在区分前两个案例和第三个情况。 的确，我们试图验证产生预测的模型 - 算法对，即产生一种具有永久性质的模式，即使这不是我们几天后我们经历的特定模式。 这种测试的失败将表明模型和/或解决方案方法是错误的。 这种方法可以应用于其他身体问题。 实际上，这些方法的初始测试将在系统上的涉及系统小于天气的涉及的系统上进行，但仍具有当前的科学兴趣。 特别是我们考虑燃烧，流体流和湍流的基本模型。