Mathematical Sciences: Pattern Formation, Turbulence and Singularities in PDEs

数学科学：偏微分方程中的模式形成、湍流和奇异性

• 批准号: 9302013
• 负责人: Alan Newell
• 金额: $ 19.5万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-06-15 至 1998-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9302013&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Pattern; Formation; Turbulence

9302013 Newell The investigator and his colleagues undertake projects aimed at gaining a better understanding of turbulence in its many manifestations and at obtaining macroscopic descriptions for patterns that serve to simplify and unify the behavior of classes of pattern forming systems in fluids and optics seemingly unrelated at the microscopic level. In the turbulence studies, the aim is to understand the origins of short-lived coherent structures and their role in causing intermittent and non-Kolmogorov-like behavior. It is important first to gain a clear picture of weak turbulence theories, the reasons for more than one Kolmogorov finite flux solution, and to understand the roles that these solutions play in producing inverse cascades and the roles that inverse cascades may play in originating collapse structures. The principal aim of the studies on pattern formation is to extend the previous far-from-onset theories to include a description of pattern behavior when the local wavenumber is driven outside of a stability band known as the Busse balloon. The results of the research so far have led to a properly regularized theory of patterns that can handle pattern singularities such as dislocations and disclinations. Moreover, the latest work suggests that all point defects of almost periodic patterns are composed of the two elementary disclinations. Patterns of an almost periodic nature arise all over the place. One sees them in the sand ripples under an advancing or receding tide, in geological formations, on the coats of animals, as fingerprints. They occur in systems driven far from equilibrium by some external stress that acts to destabilize spatially and temporally uniform states and preferentially amplify certain shapes and configurations. Geometrical symmetries, such as rotation in the plane, lead to degeneracies (the wavelength of a periodic pattern may be chosen but the direction remains undetermined) and the competition betwee n the various amplified states leads to patterns that often consist of a mosaic of almost periodic patches separated by grain boundaries, domain walls and point defects. The goal of this research is to build a macroscopic field theory for patterns that will unify and simplify the behavior of systems seemingly unrelated at the microscopic level and that can capture both the smooth (field) and singular (particle) components of the pattern.

9302013研究人员Newell和他的同事们开展的项目旨在更好地了解湍流的许多表现形式，并获得对图案的宏观描述，这些图案有助于简化和统一流体和光学中似乎在微观层面上无关的一类图案形成系统的行为。在湍流研究中，目的是了解短暂相干结构的起源，以及它们在导致间歇性和非科尔莫戈罗夫行为中的作用。重要的是，首先要清楚地了解弱湍流理论，以及产生多个Kolmogorov有限通量解的原因，并了解这些解在产生反向叶栅中所起的作用，以及反向叶栅可能在原始坍塌结构中所起的作用。关于花样形成的研究的主要目的是扩展以前尚未出现的理论，以包括当局部波数被驱动到称为Busse气球的稳定带之外时的花样行为的描述。到目前为止，这项研究的结果已经导致了一种适当的规则化的图案理论，可以处理错位和错位等图案奇点。此外，最新的工作表明，几乎周期图案的所有点缺陷都是由两个基本取向组成的。几乎周期性的模式随处可见。人们在涨潮或退潮下的沙子涟漪中，在地质构造中，在动物的皮毛上，作为指纹看到它们。它们发生在远离平衡的系统中，由某些外部应力作用，破坏空间和时间上均匀的状态，并优先放大某些形状和构型。几何对称性，如平面上的旋转，导致简并(周期图案的波长可以选择，但方向仍未确定)，各种放大状态之间的竞争导致图案，这些图案通常由由晶界、磁区壁和点缺陷隔开的几乎周期块的马赛克组成。这项研究的目标是建立一种宏观场论，它将在微观水平上统一和简化看似无关的系统的行为，并能够捕捉到模式的光滑(场)和奇异(粒子)分量。