Mathematical Sciences: Pattern Formation, Turbulence and Singularities in PDEs

小行星9302013纽韦尔 研究人员和他的同事们进行项目，旨在获得更好地了解湍流的许多表现形式，并获得宏观描述的模式，以简化和统一的行为类模式形成系统的流体和光学似乎无关的微观水平。 在湍流研究中，目的是了解短寿命相干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，一个以上的Kolmogorov有限通量解决方案的原因，并了解这些解决方案中发挥的作用，在产生逆叶栅和逆叶栅可能发挥的作用，在起源崩溃结构。 图案形成的研究的主要目的是扩展以前的远从发病理论，包括描述的图案行为时，当地波数被驱动的稳定带称为Busse气球。 到目前为止的研究结果已经导致了一个适当的正则化理论的模式，可以处理模式的奇异性，如位错和向错。 此外，最新的工作表明，所有的点缺陷几乎周期图案是由两个基本向错。 几乎是周期性的模式到处都是。 人们可以在涨潮或退潮时的沙纹中看到它们，在地质构造中，在动物的皮毛上，作为指纹。 它们发生在由一些外部应力驱动远离平衡的系统中，这些外部应力的作用是破坏空间和时间均匀状态的稳定性，并优先放大某些形状和配置。 几何对称性，如平面内的旋转，导致简并（周期性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，通常由几乎周期性的马赛克组成，由晶界，畴壁和点缺陷分开。 这项研究的目标是建立一个宏观场理论的模式，将统一和简化系统的行为，似乎无关的微观水平，可以捕捉到光滑（场）和奇异（粒子）的模式组件。

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.