Approximation of the Global Attractors of Evolution Equations

9706903 M. Jolly 摘要 这个项目关注某些耗散物理系统的长期行为。 其中一个主要组成部分是试图找到全球吸引插值手段，而不是直接数值解的初始值问题。 这将扩展我们以前的工作，使用泰勒展开在复杂的时间在相空间中的一个单一的点。 新的方法将使用几个点，通常是那些由独立计算或实验数据提供的解决方案。 主要的数学工具，在这方面的努力将是Nevalinna-Pick插值。 这将被测试的系统包括2-D Navier-Stokes（NS），Kuramoto-Sivashinsky（KS）Lorenz方程以及简单的地转模式。 另一个主要组成部分是计算不变流形到任意精度。 我们将使用2-D（不）稳定流形的可视化来帮助理解3-D相空间中某些全局分叉背后的几何机制。 这涉及到计算全局流形的主要部分。 其他应用只需要沿沿着特定轨迹计算流形。 在一种情况下，一个中心流形将被视为以这种方式来计算有界的解决方案，椭圆型偏微分方程（PDE）在一个无限的圆柱。 在另一种情况下，KS方程的相空间的维度将通过将流动限制到该维度的惯性流形而有效地减少到三维。 这样的约化将允许我们研究如上所述的全局分叉。 开发的算法来计算这些流形也将被应用到集（假设是流形）的一个规定的指数增长率向后的时间为NS方程。 事实上，我们将构造这样的集合作为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了作用，从而使我们的研究圆满完成。 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，还是仅仅是暂时的。 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，人们可以预期，除非突然的外部事件发生，否则我们现在所经历的模式将或多或少地持续一段合理的时间。这不是关于准确的长期预测，而是关于确认关于做出这些预测所使用的数学模型的基本假设。 科学界做出了巨大的努力，推导出适当的数学方程，并将其离散化，以便在计算机上求解，所有这些都产生了时间函数，该函数应该描述天气的某些方面。 我们都知道这个计算的时间函数在相对较短的时间段后偏离实际天气的频率。 这个错误的主要来源尚不清楚。 是在模型本身吗？ 是计算机解中的数值近似吗？ 或者，模型和近似值都是有效的，但实际解对初始数据的微小变化非常敏感，我们只需要收紧数据和每个时间步所用算法的误差容限。 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，因为它产生了一种永久性的模式，即使它不是我们在几天后经历的特定模式。 这种测试的失败将表明模型和/或解决方案的方法是错误的。 这种方法可以应用于其他物理问题。 实际上，这些方法的初步测试将在比天气更少涉及的系统上进行，但这些系统仍然是当前科学感兴趣的。 特别是，我们考虑燃烧，流体流动和湍流的基本模型。

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.