Mathematical Sciences: Structure of Solutions to Certain Equations in the Physical Sciences

郑9703711 郑宇熙提出从流体力学（欧拉方程）、等离子体物理（Vlasov-Poisson系统）、 和液晶物理学。这些方程是它们的运动规律。 各自的物理学材料中的湍流性质和/或缺陷 以奇点和不稳定性的形式出现在 方程式郑裕彤计划利用先进的分析工具进行研究 奇异解的结构。例如，在二维空间中的可压缩欧拉方程的情况下，郑玉玺计划隔离典型的奇点（飓风，龙卷风，冲击等）。研究它们的个体结构调查结果将是一个明确的 了解最坏的可能解决方案，从而量化我们的 物理知识，并在高性能数值计算方面提供指导 一般解的计算。 郑裕彤提出要研究数学中的一些应用问题， 流体动力学领域（包括空气和水的运动）， 等离子体物理学（包括构成太阳的物质的运动，也包括90%以上的宇宙）和液晶物理学。 材料科学 科学家和工程师们用数学公式， 叫做偏微分方程来模拟运动。湍流 材料中的性质和/或缺陷以奇点的形式出现 以及方程解的不稳定性。正是这些奇点 和不稳定性，往往破坏精确的数值计算的 解决方案Yuxi Zheng计划使用最先进的分析工具， 研究奇异解的结构。 在可压缩的 气体，如空气，例如，玉溪郑计划隔离典型的 奇点（飓风、龙卷风、冲击等）并探讨其 个别结构。调查结果将是一个明确的 了解最坏的可能解决方案，从而量化我们的 物理知识，并在高性能数值计算方面提供指导 一般解的计算。

Zheng 9703711 Yuxi Zheng proposes to study some nonlinear partial differential equations from fluid dynamics (Euler equations), plasma physics (Vlasov-Poisson system), and liquid crystal physics. These equations are the laws of motion of their respective physics. The turbulent nature and/or defects in the materials show up in the form of singularities and instabilities in the solutions of the equations. Yuxi Zheng plans to use advanced analytical tools to study the structures of the singular solutions. In the case of compressible Euler equations in two space dimensions, 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, and thereby quantify our knowledge of the physics and offer guidance in high-performance numerical computations of general solutions. Yuxi Zheng proposes to study some applied mathematical problems in the fields of fluid dynamics (which includes motion of the air and water), plasma physics (which includes motion of the material that makes up the Sun, and also more than 90% of the universe), and liquid crystal physics in material science. Scientists and engineers have used mathematical equations, called partial differential equations, to model the motions. The turbulent nature and/or defects in the materials show up in the form of singularities and instabilities in the solutions of the equations. 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, and thereby quantify our knowledge of the physics and offer guidance in high-performance numerical computations of general solutions.