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

郑9703711郑玉玺建议从流体力学(欧拉方程)、等离子体物理(弗拉索夫-泊松系统)和液晶物理来研究一些非线性偏微分方程组。这些方程是它们各自的物理运动定律。材料中的湍流性质和/或缺陷在方程的解中以奇性和不稳定性的形式表现出来。郑玉玺计划使用先进的分析工具来研究奇异解的结构。例如，在两个空间维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)。并研究它们各自的结构。调查的结果将是清楚地了解最糟糕的可能解，从而量化我们对物理的知识，并在一般解的高性能数值计算中提供指导。郑玉玺建议研究流体力学(包括空气和水的运动)、等离子体物理(包括组成太阳的物质的运动，以及宇宙90%以上的物质的运动)和材料科学中的液晶物理等领域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这些运动。材料中的湍流性质和/或缺陷在方程的解中以奇性和不稳定性的形式表现出来。正是这些奇点和不稳定性经常破坏解的精确数值计算。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。调查的结果将是清楚地了解最糟糕的可能解，从而量化我们对物理的知识，并在一般解的高性能数值计算中提供指导。

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.