Analysis of Equations in the Physical, Material, and Life Sciences

郑玉玺为NSF提案DMS-0305114PI(标题：物理、材料和生命科学中的方程分析)建议研究无粘流模型的欧拉方程，合金相场模型中的Cahn-Hilliard和Ginzburg-Landau方程，液晶模型的非线性变分波动方程，等离子体物理中的薛定谔-泊松方程和弗拉索夫-泊松方程，以及分子生物学中的蛋白质折叠问题。他的目标是更好地理解和简化复杂性，包括降低蛋白质折叠问题的复杂性，溶质对提高合金强度的影响，以及空气、水和液晶中奇点的形成机制。这些方法包括硬分析、软分析和渐近分析、数值计算和数学建模技术。建议研究的数学问题对于理解各自的学科领域都是基本的。例如，在蛋白质折叠问题中寻找三点之间距离的测量是为了大幅减少能量势的巨大局部极小值的数量，从而将复杂性带到可理解的水平。在合金相场模型中提出的问题是为控制溶质在强化合金中的作用提供定量和定性的基础。从薛定谔-泊松方程到弗拉索夫-泊松方程的极限的理论问题是一个在整体理解物质和数学模型中具有重要意义的一致性问题。对这些数学问题的研究将使人们对合金、液体、气体、等离子体、液晶和生物材料有新的认识，这些对促进蛋白质工程、药物设计、固体溶液硬化、航空航天工程、机器人设计、节能设备等许多工程科学的发展至关重要；(2)为研究生或博士后研究人员提供高级培训；(3)加强数学、材料研究、物理、生物化学、分子生物学和其他生命科学之间的合作和交叉培训，从而为在促进研究的同时培养这一广泛领域的学生奠定基础。郑玉玺建议研究流体力学(包括空气和水的运动)、合金建模、等离子体物理、分子生物学中的蛋白质折叠和材料科学中的液晶物理中的一些应用数学问题。科学家和工程师使用被称为偏微分方程式的数学方程来模拟运动或进化。物质的湍流性质和/或缺陷以及生命的复杂性，表现为方程解的奇异性和不稳定性，或者表现在方程本身的复杂性上。在蛋白质折叠问题中，需要对方程本身进行数学简化，以便计算机进行实时数值模拟。在方程式非常简单的所有其他情况下，正是这些奇点和不稳定性经常破坏解的准确数值计算。郑玉玺计划使用最先进的分析工具来研究奇异解的结构。例如，在可压缩气体(如空气)的情况下，郑玉玺计划隔离典型的奇点(飓风、龙卷风、冲击等)。并研究它们各自的结构。研究的结果将是清楚地了解最糟糕的可能的解，或大大降低复杂性，从而量化我们的物理知识，并为一般解的高性能数值计算提供指导。这里的成功将影响到合金、液体、气体、等离子体、液晶和生物材料等科学领域，并为蛋白质工程、药物设计、固体溶液硬化、航空航天工程、机器人设计、能效设备等许多工程科学的进步提供关键知识。此外，这里的成功将为研究生和博士后研究人员提供高级培训，并加强数学、材料研究、物理、生物化学、分子生物学和其他生命科学之间的合作和交叉培养，从而为在促进研究的同时培养这一广泛领域的学生奠定基础。

Abstract for NSF Proposal DMS-0305114PI: Yuxi Zheng(Title: Analysis of Equations in the Physical, Material, and Life Sciences)Yuxi Zheng proposes to study the Euler equations modeling inviscid fluids, Cahn-Hilliard and Ginzburg-Landau equations in phase-field modeling of alloys, nonlinear variational wave equations modeling liquid crystals, Schrodinger-Poisson and Vlasov-Poisson equations in plasma physics, and the protein folding problem in molecular biology. His objectives are to gain both better understanding and simplifications of complexity, which include complexity reduction for the protein folding problem, effect of solutes on the enhancement of strength of alloys, and mechanism of singularity formation in air, water, and liquid crystals. The methods include hard, soft, and asymptotic analysis, numerical computation, and techniques of mathematical modeling. The mathematical issues proposed to study are all fundamental for the understanding of the respective subject areas. For instance, the search for a measurement of distance between three points in the protein folding problem is to reduce drastically the huge number of local minima of the energy potential and thus bring the complexity to a comprehensible level. The issues proposed in the phase-field model of alloys is to provide the quantitative as well as qualitative foundation for manipulating the effect of solutes in strengthening the alloys. The theoretical issue regarding the limit from Schrodinger-Poisson to Vlasov-Poisson equations is a consistency issue of great importance in the overall understanding of matter and mathematical modeling. The investigation of these mathematical issues will(1) yield new understanding regarding alloys, liquid, gases, plasmas, liquid crystals, and bio-materials, which are critical for the advancement of many engineering sciences such as protein-engineering, drug designing, solid solution hardening, aerospace engineering, robot designing, energy efficient devices, etc.; (2) provide advanced training for graduate students or postdoctoral researchers; (3) enhance collaboration and cross training of faculties between mathematics, material research, physics, biochemistry, molecular biology, and other life sciences, thereby establish a foundation for training students in this broad area while promoting research. Yuxi Zheng proposes to study some applied mathematical problems in fluid dynamics (which includes the motion of air and water), modeling of alloys, plasma physics, protein folding in molecular biology, and liquid crystal physics in material science. Scientists and engineers have used mathematical equations, called partial differential equations, to model motions or evolution. The turbulent nature and/or defects in the materials and the complexity of life show up in the form of singularities and instabilities in the solutions of the equations or in the complexity of the equations themselves. In the protein folding problem, the equations themselves need to be mathematically simplified for a computer to do real time numerical simulation. In all the other cases, where the equations are quite simple, it is these singularities and instabilities that often spoil accurate numerical computations of the solutions. Yuxi Zheng plans to use the state of the art analytical tools to study the structures of the singular solutions. In the case of a compressible gas such as air, for example, Yuxi Zheng plans to isolate typical singularities (hurricanes, tornadoes, shocks, etc.) and investigate their individual structures. The result of the investigation will be a clear understanding of the worst possible solutions, or drastic reduction of complexity, and thereby quantify our knowledge of the physics and offer guidance in high-performance numerical computations of general solutions. The success here will influence scientific areas such as alloys, liquid, gases, plasmas, liquid crystals, and bio-materials, and provide critical knowledge for the advancement of many engineering sciences such as protein-engineering, drug designing, solid solution hardening, aerospace engineering, robot designing, energy efficient devices, etc. In addition, the success here will provide advanced training for graduate students and postdoctoral researchers and enhance collaboration and cross training of faculties between mathematics, material research, physics, biochemistry, molecular biology, and other life sciences, thereby establish a foundation for training students in this broad area while promoting research.