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和趋化性的几种基本模型。这些模型是：描述流体动力学现象的Navier-Stokes和Euler方程； 聚合物方程，该方程对非相互作用的聚合物链进行建模；描述气象现象的二维准地藻（2DQG）方程；最后，阻尼的Boussinesq系统，该系统模拟了浅水中水波的传播。该项目的一部分涉及非线性偏微分方程的中心问题，即规律性，湍流形成以及可能的明确解决方案的构建。具体而言，主要研究者关注的是为Euler和Platak-keller-segel方程建造特殊解决方案（后者是趋化现象的众所周知的模型），或者在另一个方向上，确定这种溶液不存在。 显然，这种结构在物理上是有意义的。该提案的主要部分与解决Navier-Stokes方程的稳定性有关。与其他三种流体模型有关的工作：聚合物，2DQG和BousSinesQ方程有望产生有关解决方案的长期行为的新信息。该项目中考虑的所有模型都有可能进行有趣应用。主要研究者对这些模型的兴趣源于与过去相比，特别是她能够与其他科学家进行跨学科互动的可能性。在接下来的讨论中，讨论的重点是上面提到的两个主要模型：Navier-Stoke方程和聚合方程。 关于Navier-Stokes方程的主要吸引力是，作为粘性流的模型，它用于研究血流。希望是，对方程式的广泛理论理解将最终导致测试真实流动行为的能力，并可以预测其长期行为。通过将Navier-Stokes方程与一个方程式耦合，以描述粒子位置的概率密度函数的时间演变来获得原始形式的聚合物方程。聚合物是一种由大分子质量组成的物质，该物质由通过共价化学键连接的重复结构单元（单体）组成，这些键是由于原子之间的电子共享而存在的。由共同电子引起的吸引 - 抑制稳定性是共价键的特征。赫尔曼·史丁格（Hermann Staudinger）在1920年的一份开创性和有争议的论文中引入了共价纽带之间共价键合的想法（诺贝尔（Nobel in Chemistry），《化学上的诺贝尔奖》，1953年）。解释非相互作用聚合物链的最简单模型是所谓的哑铃模型。哑铃由两个由弹性弹簧连接的珠子组成。 可以想象，在这个模型中，珠子代表原子，而弹性弹簧则起共价键的作用。这个简单的模型是该项目首先要理解的。在这里再次，压力是对哑剧经历的运动的长期行为。本提案中考虑的所有问题都可以用作与本科生和博士学位的工作的基础。项目。