Collaborative Research: Study of Turbulence in Physical Systems Through Complex Singularities and Determining Modes

该项目将最近引入的两个演化方程应用于流体动力湍流的基本问题。第一个方程是关于一类Navier-Stokes方程的确定模。一组低模式确定，如果对于全局吸引子上的任何解，该解的所有时间的高模式只能从低模式产生(对于所有时间)。将这种关系从实数推广到低模的有界函数的Banach空间，得到了一种决定性的形式，一个全局Lipschitz的发展方程，并且是耗散的。原微分方程解的全局吸引子包含在确定形式的长时间行为中。这为惯性形式理论提供了另一种选择，而惯性形式理论并不适用于Navier-Stokes方程。另一个演化方程是空间变量的解析半径。我们将使用这个方程来研究整体吸引子上的解的解析域及其对流体流动的一些方面的影响，如间歇性。大尺度的流体运动，例如在海洋和大气中，当局部和短时间观察时，显示出不稳定的，看似不可理解的(湍流)特征。然而，随着时间的推移，出现了一致的全球统计模式。这项研究的目的是对这种模式的性质和起源以及偏离这种模式的情况有一个严格的了解，以期实现海洋建模和天气预报等各种应用。例如，天气预报是通过求解一大组方程来实现的，这些方程对一段时间内的大气状态进行建模。为了计算未来的状态，需要输入大气的当前状态。由于对系统状态的测量总是有限的，因此该输入不可避免地包含一些误差。系统对这种误差的固有敏感性意味着，随着时间的推移，计算结果会偏离实际情况。数据同化技术不仅使用系统在单个时刻的有限测量，而且使用过去较长时间段的有限测量，以得出更准确的输入。这个项目应用了最新的数学发现来开发这项技术的新变体。这将导致新的算法，这些算法将由研究团队实施和测试。研究生将在这一合作努力中发挥重要作用，从而为培养STEM学科的未来一代科学家服务。

This project applies two recently introduced evolution equations to fundamental issues in hydrodynamic turbulence. The first equation is for the determining modes of a differential equation of Navier-Stokes type. A set of low modes is determining, if for any solution on the global attractor, the high modes of the solution for all time, can be produced from only the low modes (for all time). Extending this relationship to the Banach space of bounded functions from the reals into the low modes leads to a determining form, an evolution equation that is globally Lipschitz, and dissipative. The global attractor of the original differential equation is contained in the long time behavior of the determining form. This provides an alternative to the theory of inertial forms which is not known to be applicable to the Navier-Stokes equations. The other evolution equation is for the radius of analyticity in the spatial variable. We will use this equation to study the domain of analyticity for solutions on the global attractor and its effect on aspects of fluid flow such as intermittency.Fluid motions at large scales, for instance in ocean and atmosphere, display erratic, seemingly incomprehensible (turbulent) features when viewed locally and for short times. However, over time, consistent global statistical patterns emerge. The purpose of this research is to develop a rigorous understanding of the nature and origin of such patterns, and of deviations from it, with a view towards diverse applications such as ocean modeling and weather forecasting. For instance, weather forecasting is done by solving a large system of equations which model the state of the atmosphere over a period of time.The current state of the atmosphere is needed as input in order to compute the state in the future. As there are always limited measurements of the state of the system, this input inevitably contains some error. The inherent sensitivity of the system to such an error means that as time goes on the computed solution diverges from reality. The technique of data assimilation uses the limited measurements of the system at not just a single moment of time, but rather over an extended time period in the past, to arrive at a more accurate input. This project applies recent mathematical discoveries to develop new variations on this technique. It will lead to new algorithms that will be implemented and tested by the research team. Graduate students will play an essential role in this collaborative effort, thus serving to train future generation of scientists in the STEM disciplines.