COLLABORATIVE: Mathematical Sciences: Dynamics of Interfaces and Phase Transition

9622791 Alikakos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体的数学模型。数学问题解决了一般问题：非线性系统如何松弛到平衡？界面和缺陷是如何形成的？它们又是如何传播的？这些问题出现在扩散和尖锐界面模型中。我们在相界面动力学中也遇到了一些有趣的问题，这些问题与几何中的有趣问题密切相关。我们打算获得一些新的模型和经典的相变模型所预测的相态的定性行为的信息。我们将考虑等温固-固转变，这种转变发生时，每种物质的总量都没有变化，我们也将研究液-固转变，其中热方程与序参数相耦合。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。然后将其简化为有限维动力系统通常是可能的。在这些研究中，我们花了大量的精力来识别和理解潜在的有限维动力学。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原因。一般来说，这是二阶效应。然而，当其他力相互平衡时，表面张力可以成为决定因素。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：火箭一旦脱离地球的引力，发动机就会像燃料耗尽一样关闭。最终，他们意识到，在太空中，由于表面张力的影响，燃料箱中的燃料形成了5个球形，许多液滴远离容器壁。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华氏度的临界温度附近共存）。了解某些相的混合方式对确定材料的性质是很重要的。超导性是我们的工作可以应用的一个很好的例子。为了理解这一切，原则上，我们可以从物理学的基本方程开始。这种方法的问题在于它的复杂性和数值模拟的难度。为了解决这个问题，科学家们提出了特别的、在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。它旨在证明一些简单的数学模型表现出丰富的行为，使它们能够描述和预测所观察到的现象。＊＊＊

9622791 Alikakos 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 s pherical 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 . ***