Mathematical Sciences: Pattern Formation, Turbulence and Singularities in PDEs

研究者和他的同事们进行的项目旨在更好地理解湍流的许多表现形式，并获得模式的宏观描述，这些模式有助于简化和统一流体和光学中看似在微观层面上无关的模式形成系统的行为。在湍流研究中，目的是了解短时间连贯结构的起源及其在引起间歇性和非柯尔莫哥洛夫行为中的作用。重要的是，首先要清楚地了解弱湍流理论，了解存在多个Kolmogorov有限通量解的原因，并了解这些解在产生反级联中所起的作用，以及反级联在起源坍塌结构中可能起的作用。模式形成研究的主要目的是扩展以前的远起始理论，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模式行为的描述。到目前为止，研究的结果已经导致了一个适当的正则化的模式理论，可以处理模式奇点，如位错和偏差。此外，最新的研究表明，几乎周期性图案的所有点缺陷都是由这两个基本缺陷组成的。几乎周期性的模式随处可见。人们可以在涨潮或退潮时的沙波纹中看到它们，在地质构造中看到它们，在动物的皮毛上看到它们，就像指纹一样。它们发生在远离平衡的系统中，这些系统受到一些外部压力的驱动，这些外部压力会破坏空间和时间上的均匀状态，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，但方向仍然不确定），各种放大状态之间的竞争导致图案通常由由晶界，畴壁和点缺陷分隔的几乎周期性斑块的马赛克组成。本研究的目标是为模式建立一个宏观场论，统一和简化在微观层面上看似无关的系统的行为，并可以捕获模式的光滑（场）和奇异（粒子）成分。

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.