COLLABORATIVE: Mathematical Sciences: Dynamics of Interfaces and Phase Transitions

小行星9622785 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统的数学模型。 数学问题解决一般问题：非线性系统如何放松到平衡？ 接口和缺陷是如何形成的，它们又是如何传播的？ 这些问题都出现扩散和尖锐的界面模型。 我们也遇到有趣的问题，在相界面动力学，这是密切相关的有趣的问题，几何。 我们打算获得一些新的模型以及经典的相变模型预测的相态的定性行为的信息。 我们将考虑等温固-固转变发生在没有变化的总量的每一个物种，我们也将研究液-固转变的热方程耦合到该序参数。在自然情况下，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，减少到有限维动力系统往往是可能的。 在这些研究中，我们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，这是一个二阶效应。 然而，当其他力相互平衡时，表面张力可以成为决定因素。 下面是一个例子：在太空计划的第一阶段，科学家们感到困惑的是，一旦火箭离开地球的引力，发动机就会关闭，好像燃料耗尽一样。 最终，他们意识到，在太空中，由于表面张力的影响，油箱中的燃料形成了球形，许多斑点远离容器壁。 我们对相变现象感兴趣，例如那些涉及冰和水的现象（同一物质的不同相可以在32 F的临界温度附近共存）。 了解某些相混合的方式对于确定材料的性质是很重要的。超导性是我们的工作适用的一个很好的例子。 为了理解这一切，原则上可以从物理学的基本方程开始。这种方法的问题是遇到的巨大复杂性和数值模拟的困难。为了解决这个问题，科学家们提出了一些特别的、现象学上令人信服的模型，这些模型具有非常简单的优点。 我们的工作是定性的。它旨在建立这些简单的数学模型中的一些表现出丰富的行为，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 . ***