Harmonic analysis and applications to geometric PDE's

调和分析及其在几何偏微分方程中的应用

• 批准号: 0300511
• 负责人: Atanas Stefanov
• 金额: $ 9.8万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-05-15 至 2007-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0300511&HistoricalAwards=false
• 关键词: Harmonic; analysis; applications; geometric; PDE

PI: Atanas G. StefanovDMS-0300511ABSTRACT:The proposal deals with several problems arising in geometry and in the theory of electromagnetism. The main goal is to study analytically the behavior of the solutions of the partial differential equations, describing these processes. The PI will concentrate on the study of the following equations: the Wave map problem, the Schroedinger map problem and the Yang-Mills fields equations. Following the fundamental physical principle that every physical system is trying to minimize its energy, one form and studies the solutions to these nonlinear partial differential equations, which arise as the minimizers of the corresponding energy functionals. The questions for existence, uniqueness and stability for these equations have been studied extensively in various settings. However to conform to a more natural physical situation, one needs to consider low regularity data. Stefanov will concentrate on showing regularity for solutions with critical degree of regularity. That is, one requires the least possible amount of smoothness in order to show that the solution exists globally in time. In the physically interesting case of small energy data (modeling systems that are very close to their equilibrium), the proposal will address the question of stability, i.e. whether the system will remain close to its initial state or will eventually develop some singularities. Stefanov proposes to study these problems, by using a variety of techniques including frequency analysis and gauge theoretic methods. A concurrent aim of the project is to make these methods more readily applicable to other problems. The wave equation models the propagation of different kind of waves in materials. Nonlinear models of conservative type arise in the study of vibrating systems and semiconductors.The nonlinear Schroedinger equation describes the propagation of a laser beam in a medium whose index of refraction is sensitive to the wave amplitude. Better theoretical understanding of the time evolution of these models will allow one to better predict the physical behaviour of such systems. For example, if a physical system of the kind described above, is stable (rigid) with respect to small perturbations, one could afford to have certain amount of noise/impurities in thesystem. The analytical study of the equations governing such processes will lead to a better understanding of the corresponding physical phenomenon.

PI：Atanas G. Stefanovdms-0300511abtract：该提案涉及几何学和电磁理论中产生的几个问题。主要目标是分析研究部分微分方程的解决方案的行为，描述这些过程。 PI将集中于以下方程的研究：波图问题，Schroedinger地图问题和Yang-Mills字段方程。遵循每个物理系统试图最大程度地减少其能量的基本原则，一种形式并研究了这些非线性部分偏微分方程的解决方案，这些方程是作为相应能量功能的最小化器所产生的。在各种情况下，已经对这些方程式的存在，独特性和稳定性进行了广泛研究。但是，要符合更自然的身体状况，需要考虑较低的规律性数据。 Stefanov将集中精力显示具有临界程度的固定性的规律性。也就是说，一个人需要最少的平滑度，以表明该解决方案在全球范围内存在。在小型能量数据的物理上有趣的情况下（非常接近其平衡的建模系统），该提案将解决稳定问题，即该系统是否将保持接近其初始状态或最终将发展一些奇异性。 Stefanov建议通过使用各种技术（包括频率分析和计量理论方法）来研究这些问题。该项目的同时目的是使这些方法更容易适用于其他问题。 波方程模拟了材料中不同类型波的传播。保守类型的非线性模型在振动系统和半导体的研究中出现。非线性Schroedinger方程描述了激光束在介质中的激光束传播，其折射率对波振幅敏感。对这些模型的时间演变的更好理论理解将使人们更好地预测此类系统的身体行为。例如，如果上述类型的物理系统相对于小扰动是稳定的（刚性），则可以在该系统中具有一定程度的噪声/杂质。对此类过程的方程式的分析研究将使人们更好地理解相应的物理现象。