Singular Solutions to Certain Equations in the Physical Sciences

从流体力学和液晶物理的角度研究非线性偏微分方程。这些方程是它们各自的物理运动定律。材料的湍流性质和/或缺陷使得方程的解奇异、不稳定和难以计算。计划使用先进的分析工具来研究奇异解的结构。以液晶波动方程为例，计划是将其作为基本线性波动方程的非线性推广的最简单例子之一进行研究。从其优美的形式可以看出，该方程在未来将有许多应用。其解的奇点最近被首席研究员和他的合作者证明是存在的。正是这些奇点阻碍了解的存在性、唯一性和稳定性的一般理论的建立。计划对奇异点进行详细的估计和分析，改进现有的紧性论证，形成解的存在性理论。调查的结果将是对最坏可能解的清晰理解，从而量化我们的物理知识，并为一般解的数值计算提供指导。该研究将涉及流体动力学（包括空气和水的运动）和材料科学中液晶物理领域的一些应用数学问题的研究。科学家和工程师使用偏微分方程来模拟这些运动。材料的湍流性质和/或缺陷以方程解的奇异性和不稳定性的形式表现出来。正是这些奇异性和不稳定性常常破坏解的精确数值计算。计划使用最先进的分析工具来研究奇异解的结构。以空气等可压缩气体为例，首席研究员计划分离出典型的奇点（飓风、龙卷风、冲击等），并研究它们各自的结构。调查的结果将是对最坏可能解的清晰理解，从而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。

0071858ZhengIt is proposed to study some nonlinear partial differential equations from fluid dynamics and liquid crystal physics. These equations are the laws of motion of their respective physics. The turbulent nature and/or defects in the materials make the solutions of the equations singular, unstable, and hard to calculate. It is planned to use advanced analytical tools to study the structures of the singular solutions. In the case of the liquid crystal wave equation, for example, the plan is to study it as one of the simplest examples of nonlinear generalizations of the basic linear wave equation. It can be seen from its elegant form that there will be many applications of the equation in the future. Singularities of its solutions have been shown to exist recently by the principal investigator and his collaborators. It is these singularities that block the establishment of a general theory of existence, uniqueness, and stability of solutions. It is planned to do detailed estimates and analysis on the singularities and improve current compactness arguments to form an existence theory of solutions. 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 numerical computations of general solutions.The research will involve the study of some applied mathematical problems in the fields of fluid dynamics (which includes motion of the air and water) 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. It is planned 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, the principal investigator 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.