Analysis of Equations in the Applied Sciences

研究了无粘流体的欧拉方程，合金相场模型的Cahn-Hilliard和Ginzburg-Landau方程，液晶模型的非线性变分波方程。他的目标是更好地理解和简化复杂性（包括多维冲击反射问题的复杂性降低），溶质对合金强度增强的影响，以及空气、水和液晶中奇点形成的机制。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。数学问题是理解各自学科领域的基础。例如，在多维冲击反射问题中寻求模型，以减少可能的湍流，从而使复杂性达到可理解的水平，并解决冯·诺伊曼悖论（例如，规则冲击反射和马赫反射之间的矛盾边界）。合金相场模型的问题是为控制溶质强化合金提供定量和定性的基础。对这些数学问题的研究(1)产生了对合金、液体、气体和液晶的新理解，这些对许多工程科学的进步至关重要，如固溶体硬化、航空航天工程、机器人设计和节能设备；(2)为研究生或博士后研究人员提供高级培训；(3)加强数学、材料研究和物理之间的合作和交叉训练，从而为培养这一广泛领域的学生奠定基础。研究流体动力学（包括空气和水的运动）、合金建模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偏微分方程来模拟运动和进化。流体流动的湍流性质、材料的缺陷和生命的复杂性，都以奇异性和不稳定性的形式出现在方程的解中，或者在方程本身的复杂性中出现。在方程非常简单的情况下，正是这些奇异性和不稳定性常常破坏解的精确数值计算。研究者使用分析数学工具来研究这些奇异解的结构。以空气这样的可压缩气体为例，他分离出典型的奇点（飓风、龙卷风、冲击等），并研究它们各自的结构。调查的结果是对最坏的可能——最奇异的——解有了更清晰的理解，或者是复杂性的大幅降低，这将量化我们的物理知识，并为通用解的高性能数值计算提供指导。这些结果影响了诸如天气预报、合金、液体、气体和液晶等科学领域，并为许多工程科学的进步提供了关键知识，如固溶体硬化、航空航天工程、机器人设计和节能设备。此外，该项目为研究生和博士后研究人员提供了高级培训机会，并加强了数学，材料研究和物理之间的合作和交叉训练，为培养这一广泛领域的学生奠定了基础。

ZhengDMS-0603859 The investigator studies the Euler equations modelinginviscid fluids, Cahn-Hilliard and Ginzburg-Landau equations inphase-field modeling of alloys, and nonlinear variational waveequations modeling liquid crystals. His objectives are to gainboth better understanding and simplifications of complexity(which includes complexity reduction for multi-dimensional shockreflection problems), of the effect of solutes on the enhancementof strength of alloys, and of the mechanism of singularityformation in air, water, and liquid crystals. The methodsinclude hard, soft, and asymptotic analysis, numericalcomputation, and techniques of mathematical modeling. Themathematical issues are fundamental for the understanding of therespective subject areas. For instance, a model is sought inmulti-dimensional shock reflection problems to reduce possibleturbulence and thus bring the complexity to a comprehensiblelevel and settle the von Neumann paradoxes (e.g., the paradoxicalboundary between regular shock reflection and Mach reflection). The issue in the phase-field model of alloys is to provide aquantitative as well as qualitative foundation for manipulatingsolutes to strengthen the alloys. Study of these mathematicalissues (1) yields new understanding regarding alloys, liquids,gases, and liquid crystals, which are critical for theadvancement of many engineering sciences such as solid solutionhardening, aerospace engineering, robot designing, and energyefficient devices; (2) provides advanced training for graduatestudents or postdoctoral researchers; (3) enhances collaborationand cross-training between mathematics, material research, andphysics, thereby establishing a foundation for training studentsin this broad area. The investigator studies some applied mathematical problemsin fluid dynamics (which includes the motion of air and water),modeling of alloys, and liquid crystal physics in materialscience. Scientists and engineers have used mathematicalequations, called partial differential equations, to modelmotions or evolution. The turbulent nature of fluid flows,defects in materials, and the complexity of life show up in theform of singularities and instabilities in the solutions of theequations or in the complexity of the equations themselves. Incases where the equations are quite simple, it is thesesingularities and instabilities that often spoil accuratenumerical computations of the solutions. The investigator usesanalytical mathematical tools to study the structures of thesingular solutions. In the case of a compressible gas such asair, for example, he isolates typical singularities (hurricanes,tornadoes, shocks, etc.) and investigates their individualstructures. The result of the investigation is a clearerunderstanding of the worst possible -- most singular --solutions, or a drastic reduction of complexity, which quantifiesour knowledge of the physics and offers guidance inhigh-performance numerical computations of general solutions. Such results influence scientific areas such as weatherforecasting, alloys, liquid, gases, and liquid crystals, andprovide critical knowledge for the advancement of manyengineering sciences such as solid solution hardening, aerospaceengineering, robot design, and energy-efficient devices. Inaddition, the project provides opportunities for advancedtraining for graduate students and postdoctoral researchers andenhances collaboration and cross-training between mathematics,material research, and physics, creating a foundation fortraining students in this broad area.