Questions on Diffusive Phenomena

这个项目考虑了流体动力学中的几个基本模型和一个来自趋化性的模型。这些模型是：描述流体力学现象的Navier-Stokes和Euler方程；模拟非相互作用的聚合物链的聚合物方程；描述气象现象的二维准地转(2DQG)方程；以及最后是模拟浅水中水波传播的阻尼布西内斯克系统。该项目的一部分涉及非线性偏微分方程组的中心问题，即正则性、湍流的形成和可能的显式解的构造。具体地说，主要研究人员关注欧拉方程和普拉塔克-凯勒-赛格尔方程(后者是一个著名的趋化现象模型)的特解的构造，或者在另一个方向上，建立这样的解不可能存在。很明显，这种结构在物理上是有意义的。该提案的一个核心部分是关于Navier-Stokes方程解的稳定性。与其他三个流体模型：聚合物、2DQG和Boussinesq方程相关的工作有望获得有关溶液长期行为的新信息。本项目中考虑的所有模型都具有有趣的应用潜力。首席研究人员对这些模型的兴趣源于她有可能从事比过去更实用的方面的工作，特别是能够与其他科学家进行跨学科互动。在接下来的讨论中，重点讨论了上面提到的两个主要模型：Navier-Stoke方程和聚合物方程。关于纳维尔-斯托克斯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。人们希望，对该方程的广泛理论理解将最终导致测试真实流动行为的能力，或许还能预测其长期行为。通过将Navier-Stokes方程与描述粒子位置的概率密度函数的时间演化的方程耦合，得到了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，这些大分子质量由重复的结构单元(单体)组成，这些结构单元(单体)通过共价化学键相连，这些结构单元是由于原子之间共享电子而存在的。由公共电子引起的吸引-排斥稳定性是共价键的特征。长原子链之间的共价键的概念是在1920年Hermann Staudinger(诺贝尔化学奖得主)的一篇开创性和有争议的论文中提出的。解释不相互作用的高分子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，而弹性弹簧扮演共价键的角色。这个简单的模型是该项目首先要了解的。这里的重点又是哑铃所经历的运动的长期行为。这份提案中考虑的所有问题都可以作为本科生工作和博士项目的基础。

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.