Research on Nonlinear Partial Differential Equations in Conservation Laws and Related Applications

本项目致力于非线性偏微分方程守恒定律的数学研究及其应用，以及高效非线性技术的分析与开发。这项工作有三个相互关联的方面：(i)激波反射-衍射；（ii）多维冲击前缘的稳定性；（三）微分几何中的等距嵌入。他们统一的数学主题是混合双曲-椭圆型非线性偏微分方程理论和相关的非线性技术。在每个主题中，将提出涉及混合型偏微分方程的非线性问题，并设计相应的数学技巧来解决这些问题。因此，通过研究重要的非线性应用偏微分方程，将获得新的物理见解，有效的非线性技术将得到发展，并建立正确的数学框架来提出和讨论这些问题。这项工作还将导致非线性技术的分析和发展，这些技术将在更广泛的重要非线性偏微分方程中得到应用。虽然双曲型非线性偏微分方程（如描述波的传播）和椭圆型非线性偏微分方程(如：而对于固体变形，混合双曲-椭圆型方程的数学研究和相关应用还很不发达。当描述固体附近接近声速的气体流动时，就会出现这种类型的问题。例如飞机机翼和涡轮部件。本奖项所支持的工作将支持涉及混合型偏微分方程及其相应数学技术的非线性问题的研究。这将导致对混合双曲-椭圆型非线性现象的更深入的理解，并将为应用提供有效的非线性方法和理论。考虑到应用的丰富性和范围，本研究项目将通过研究几个重要的混合型非线性偏微分方程及其相关应用，并为其进一步研究开发方法和一套非线性技术，产生广泛的影响。该项目将(i)产生对气体、流体和几何数学的新理解，这对空气动力学、工业气体处理、材料科学、医学成像和环境科学至关重要；（ii）为研究生、博士后助理和其他初级研究人员提供高级培训，包括一些来自代表性不足群体的成员。此外，由于高效数值格式的设计取决于对底层数学结构和模式的理解，因此该项目的成功将有助于数值分析。

This project is focused on mathematical research on nonlinear partial differential equations in conservation laws and related applications, along with the analysis and development of efficient nonlinear techniques. The work has three interrelated aspects: (i) shock reflection-diffraction; (ii) stability of multidimensional shock-fronts; and (iii) isometric embedding in differential geometry. Their unifying mathematical theme is the theory of nonlinear partial differential equations of mixed hyperbolic-elliptic type and related nonlinear techniques. In each topic, nonlinear problems involving partial differential equations of mixed type will be posed and corresponding mathematical techniques for their solution will be devised. As a result, new physical insights will be gained by investigating important nonlinear applied partial differential equations, effective nonlinear techniques will be developed, and the correct mathematical frameworks in which to pose and discuss these problems will be established. The work will also lead to the analysis and development of nonlinear techniques which will have applications in broader classes of important nonlinear partial differential equations.While great progress has been made for nonlinear partial differential equations of hyperbolic type (e.g. for the description of wave propagation) and of elliptic type (e.g.. for deformations of solid bodies) respectively, the mathematical study of such equations of mixed hyperbolic-elliptic type and related applications is much less developed. Problems of this type occur when the behavior of gas flow near the speed of sound in the vicinity of solid objects is described. Examples include aircraft wings and turbine components. The work supported by this award will support research on nonlinear problems involving partial differential equations of mixed type and corresponding mathematical techniques. It will lead to a deeper understanding of nonlinear phenomena of mixed hyperbolic-elliptic type and will provide efficient nonlinear methods and theories for applications. Given the richness and range of applications, this research project will have a broad impact by investigating several important nonlinear partial differential equations of mixed type and related applications and by developing methodology and a set of nonlinear techniques for their further study. The project will (i) yield new understanding of the mathematics of gases, fluids, and geometry, which is critical for aerodynamics, industrial gas processing, materials science, medical imaging, and environmental science; (ii) provide advanced training for graduate students, post-doctoral associates, and other junior researchers, including several members from underrepresented groups. Furthermore, since the design of efficient numerical schemes hinges on the understanding of the underlying mathematical structure and pattern, success with this project will be useful to numerical analysis.