COLLABORATIVE: Mathematical Sciences: Dynamics of Interfaces and Phase Transitions

9622785贝茨我们研究计划的主要目标是开发和分析承认相变或结构缺陷的连续介质的数学模型。数学问题解决的是一般问题：非线性系统如何松弛到平衡？界面和缺陷是如何形成的，它们是如何传播的？对于漫反射和锐化界面模型，都会出现这些问题。我们还在相界面动力学中遇到了有趣的问题，这些问题与几何中的有趣问题密切相关。我们打算获得由几个新的相变模型以及经典相变模型预测的相态的定性行为的信息。我们将考虑在不改变每个物种的总量的情况下发生的等温固-固转变，我们还将研究热方程与序参数的热方程耦合的液-固转变。在某些自然情况下，界面的周长或几何形状在演化过程中不会发生显著变化。然后，通常可以将其归结为有限维动力系统。我们在这些研究中的大量努力都花在了识别和理解潜在的有限维动力学上。%我们研究表面张力起作用的某些物理现象。表面张力决定了行星的形状，也决定了雨滴的形状。这也是为什么在冰上滑冰是可能的。一般来说，这是一种二阶效应。然而，当其他力量相互平衡时，表面张力可能成为决定因素。这里有一个例子：在太空计划的第一阶段，科学家们对这样一个事实感到困惑：火箭一离开地球的引力，发动机就会关闭，就像燃料耗尽一样。最终他们意识到，在太空中，由于表面张力效应，油箱中的燃料形成了球形形状，许多气泡远离容器的壁。我们对相变现象感兴趣，例如涉及冰和水的相变现象(同一物质的不同相可以在32华氏度的临界温度附近共存)。了解某些相的混合方式对于确定材料的性能很重要。超导是我们工作的一个很好的例子。要理解这一切，原则上可以从物理学的基本方程开始。这种方法的问题是人们会遇到巨大的复杂性和数值模拟的困难。为了解决这个问题，科学家们提出了特别的、现象上令人信服的模型，这些模型具有极大的简单性。我们的工作是定性的。它的目的是证明其中一些简单的数学模型展示了丰富的行为，使它们能够描述和预测所观察到的现象。***

9622785 Bates The main goal of our research program is the development and analysis of mathematical models of continua which admit phase transitions or structural defects. The mathematical problems address the general questions: How do nonlinear systems relax to equilibrium? How do interfaces and defects form and how do they propagate? These questions arise both for diffuse and sharp interface models. We also encounter interesting questions in phase interface dynamics which are closely tied to interesting questions in geometry. We intend to obtain information on the qualitative behavior of phase states predicted by several new models as well as classical modelsfor phase transition. We will consider isothermal solid-solid transitions which occur without a change in the total amount of each species and we will also study liquid-solid transformations where a heat equation is coupled to that for the order parameter. There are natural situations where the perimeter or the geometry of the interface, does not change significantly during the evolution. Then a reduction to a finite dimensional dynamical system is often possible. A good deal of our effort in these investigations is spent towards identifying and understanding the underlying finite dimensional dynamics. %%% We investigate certain physical phenomena where surface tension plays a role. Surface tension is responsible for the shape of planets, as well as for the shape of rain drops. It is also the reason why skating on ice is possible. Generally it is a second order effect. However when the other forces balance each other, surface tension can become the determining factor. Here is anexample: In the first stages of the space program scientists were puzzled by the fact that as soon as the rocket was exiting the earth's gravitiational attraction the engines would shut off as if they were running out of fuel. Eventually they realized that in space the fuel in the tank forms into sphe rical shapes due to surface tension effects with many blobs staying away from the walls of the container. We are interested in phase change phenomena , for example those involving ice and water (different phases of the same substance which can coexist near the critical temperature of 32F). Understanding the way certain phases mix is important for determinig the properties of materials. Superconductivity is a good example to which our work applies. For understanding all this, one, in principle, could start with the basic equations of physics. The problem with this approach is the enormous complexity that one encounters and the difficulty in the numerical simulation. For dealing with this scientists are proposing adhoc, phenomenologically convincing models, which have the advantage of great simplicity. Our work is of qualitative nature. It aims to establish that some of these simple mathematical models exhibit the wealth of behavior making them capable of describing and predicting the phenomena that are observed . ***