Questions on Diffusive Phenomena

关于扩散现象的问题

• 批准号: 0900909
• 负责人: Maria Schonbek
• 金额: $ 22.15万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900909&HistoricalAwards=false
• 关键词: Questions; Diffusive; Phenomena

This project considers several fundamental models arising in fluid dynamics and one from chemotaxis. These models are: the Navier-Stokes and Euler equations, which describe hydrodynamics phenomena; polymeric equations, which model noninteracting polymer chains; the two-dimensional quasi-geostrophic (2DQG) equations, which describe meteorological phenomena; and, finally, the damped Boussinesq system, which models the propagation of water waves in shallow water. Part of the project relates to the central questions for nonlinear partial differential equations, namely, regularity, formation of turbulence, and the possible construction of explicit solutions. Specifically, the principal investigator is concerned with the construction of special solutions to the Euler and Platak-Keller-Segel equations (the latter is a well-known model for chemotaxis phenomena) or, in the other direction, with establishing that such solutions cannot exist. It is clear that such constructions are physically meaningful. A central part of the proposal is concerned with the stability of solutions to the Navier-Stokes equations. The work proposed in connection with the other three fluid models: the Polymeric, the 2DQG, and the Boussinesq equations is expected to yield new information on the long-time behavior of the solutions.All of the models considered in this project have the potential for interesting applications. The principal investigator's interest in these models stems from the possibility of working on a more applied side of the subject than she has in the past, in particular, to be able to have an interdisciplinary interaction with other scientists. In what follows the discussion focuses on two of the main models mentioned above: the Navier-Stoke equations and the polymeric equations. The main attraction with respect to the Navier-Stokes equations is that, as a model for viscous flows, it is used to study, among many other things, blood flow. The hope is that a broad theoretical understanding of the equation will lead eventually to the ability to test for the behavior of the real flow, and perhaps predict its long-time behavior. The polymer equations in its original form is obtained by coupling the Navier-Stokes equations with an equation that describes the time evolution of the probability density function of the position of a particle. A polymer is a substance composed of a large molecular mass consisting of repeated structural units (monomers) connected by covalent chemical bonds, which exist due to the sharing of electrons between atoms. The attraction-repulsion stability that is caused by the common electron is what characterizes the covalent bonding. The idea of covalent bonding between long chains of atoms was introduced in a ground-breaking and controversial paper by Hermann Staudinger in 1920 (Nobel Laureate in Chemistry, 1953). The simplest model to account for noninteracting polymer chains is the so-called dumbbell model. A dumbbell consists of two beads connected by an elastic spring. One can imagine that in this model the beads represent the atoms, while the elastic spring plays the role of the covalent bond. This simple model is what the project will seek to understand first. Here again the stress is on the long-time behavior of the motion that the dumbells undergo. All the problems considered in this proposal can be used as basis for work with undergraduate students and for Ph.D. projects.

该项目考虑了流体动力学中的几个基本模型和一个趋化性模型。这些模型是：Navier-Stokes和Euler方程，描述流体力学现象;聚合物方程，模拟非相互作用的聚合物链;二维准地转（2DQG）方程，描述气象现象;最后，阻尼Boussinesq系统，模拟浅水中水波的传播。该项目的一部分涉及非线性偏微分方程的中心问题，即正则性，湍流的形成，以及显式解的可能构造。具体来说，主要研究者关注的是欧拉方程和Platak-Keller-Segel方程的特殊解的构造（后者是一个著名的趋化现象模型），或者在另一个方向上，建立这样的解不存在。 很明显，这样的结构是有物理意义的。该提案的一个核心部分是关于Navier-Stokes方程解的稳定性。这项工作提出了与其他三个流体模型：聚合物，2DQG， 而Boussinesq方程则有望提供关于解的长时间行为的新信息。本项目所考虑的所有模型都具有潜在的有趣应用。首席研究员对这些模型的兴趣源于比她过去更有可能在该主题的应用方面工作，特别是能够与其他科学家进行跨学科的互动。在下文中，讨论集中于上述两个主要模型：Navier-Stoke方程和聚合物方程。 Navier-Stokes方程的主要吸引力在于，作为粘性流动的模型，它被用来研究血液流动等许多问题。希望是，对方程的广泛理论理解最终将导致测试真实的流动行为的能力，并可能预测其长期行为。在其原始形式的聚合物方程是通过耦合的Navier-Stokes方程的方程，描述了粒子的位置的概率密度函数的时间演化。聚合物是由大分子质量组成的物质，该大分子质量由通过共价化学键连接的重复结构单元（单体）组成，该共价化学键由于原子之间共享电子而存在。由公共电子引起的吸引-排斥稳定性是共价键的特征。1920年，赫尔曼·施陶丁格（Hermann Staudinger，1953年诺贝尔化学奖获得者）在一篇开创性和有争议的论文中提出了原子长链之间共价键的想法。最简单的模型来解释非相互作用的聚合物链是所谓的哑铃模型。哑铃是由两个由弹性弹簧连接的珠子组成的。 可以想象，在这个模型中，珠子代表原子，而弹性弹簧扮演共价键的角色。这个简单的模型是项目首先要理解的。这里的重点再次放在哑铃运动的长期行为上。本建议中考虑的所有问题都可以作为与本科生和博士生一起工作的基础。项目