Problems in Dynamics of Phase Transitions and Numerical Analysis of Conversation Laws

范海涛摘要：拟议项目的长期目标是了解液/汽相变的动力学及其数学理论。液/汽相变是科学和工程中的重要现象。传统的相变动力学模型与实验结果存在较大差距。我们建议通过一种不同于传统模型的方法来研究液/汽相变动力学。该模型是一个基于两相反应流方程的源项守恒律系统。根据我们的初步研究，这种方法有望更好地符合实验。为了实现我们的目标，将追求以下特定子目标：1)推导系统的定性和定量渐近模型。2)研究了系统的基本波型及其渐近模型，如行波、激波和相边界的容许性准则、黎曼问题等。3)开发算法以对系统进行数值模拟，并开发测试问题以证明这些计算是正确的。研究与系统数值格式有关的问题4)研究系统或其简化渐近模型的适定性。预期是，在拟议的支助期结束时，将获得关于系统定性和定量行为的结果，并将进行数值模拟。系统定性和定量行为的结果将与实验进行比较。基于这些比较，可以评估建模的充分性，并且可以进一步改进建模。预计所获得的结果将有助于理解液/气相变的动力学。这项工作还将产生对这类现象进行数值模拟的工具。大多数材料有四相S：固、液、汽相和等离子体相。相变，如从水到水蒸气的变化，反之亦然，是日常生活中和我们环境中随处可见的材料的基本现象。许多制造过程，例如用于半导体制造的单个大晶体的生长，都涉及相变。液体/蒸汽相变发生在飞机穿越云层的飞行和水下爆炸过程中。在舰船的高速运动部分，如潜艇的螺旋桨，也会发生液/汽相变化，这会消耗能量，并使敌人的太阳能设备能够检测到噪音。总之，相变不仅是材料的基本科学性质，而且对工业竞争力和国家安全至关重要。拟议的项目是研究相变的演化，特别是由快速运动的物体或波引起的快速液/汽相转变。其目标是建立与实验数据相一致的有效的液/汽相变数学模型，并开发此类相变的数值模拟工具。

Haitao Fan Abstract: The long-range goal for the proposed project is to understand the dynamics of liquid/vapor phase transitions and its mathematical theories. Liquid/vapor phase transitions are important phenomena in science and engineering. Traditional models for the dynamics of phase transition, exhibit gaps when compared with experiments. We propose to investigate the dynamics of liquid/vapor phase transitions through an approach different from traditional models. The model is a system of conservation laws with source terms (the system) based on equations for two-phase reactive flows. Based our preliminary studies, this approach promises to better fit experiment. To accomplish our objective, the following specific sub-objectives will be pursued: 1) Derive qualitative and quantitative asymptotic models for the system. 2) Study issues such as basic wave patterns in the system and its asymptotic models, e.g., traveling waves, admissibility criteria for shocks and phase boundaries, and the Riemann problem. 3) Develop algorithms to carry out numerical simulations on the system and develop test problem to justify such computations. Study issues related to numerical schemes for the system 4) Study the well-posedness of the system or its simplified asymptotic models. The expectations are that, at the conclusion of the proposed period of support, results will have been obtained on qualitative and quantitative behavior of the system, and numerical simulations will have been carried out. Results on qualitative and quantitative behavior of the system will be compared with experiments. Based on these comparisons, the adequateness of the modeling can be assessed, and the modeling can be further improved. It is expected that the results obtained will facilitate understanding of the dynamics of liquid/vapor phase transitions. Tools for numerical simulations of such phenomena will also result from this work. Most materials have four phase s: solid, liquid, vapor and plasma phases. Phase transition, such as the change from water to vapor and vice versa, is fundamental phenomena of materials that occur in daily life and everywhere in our environment. Many manufacturing processes, such as the growth of a single large crystal used in semiconductor manufacturing, involve phase transitions. Liquid/vapor phase transitions happen during the flight of airplane through clouds and during underwater explosions. Liquid/vapor phase changes also occur at high speed moving parts of a vessel, such as the propeller of a submarine, which consumes energy and makes noise detectable by enemies' solar devices. In summary, phase transitions are not only fundamental scientific properties of materials but also vital to industrial competitiveness and national security. The proposed project is to study the evolution of phase changes, in particular the rapid liquid/vapor transitions induced by rapid moving body or waves. The objectives are to establish effective mathematical models for liquid/vapor transitions that agree with experimental data, and to develop numerical simulation tools for such phase transitions.