Harmonic analysis and applications to geometric PDE's

摘要：该提案涉及几何和电磁学理论中出现的几个问题。主要目的是分析地研究描述这些过程的偏微分方程解的行为。PI将集中研究以下方程：波图问题，薛定谔图问题和杨-米尔斯场方程。遵循基本的物理原理，即每个物理系统都试图最小化其能量，一个形式并研究这些非线性偏微分方程的解，这些解作为相应能量泛函的最小值而出现。这些方程的存在性、唯一性和稳定性问题在各种情况下得到了广泛的研究。然而，为了符合更自然的物理情况，需要考虑低规律性数据。Stefanov将专注于显示具有临界正则度的解的正则性。也就是说，人们需要尽可能少的平滑，以表明解决方案在时间上全局存在。在小能量数据（非常接近其平衡的建模系统）的物理有趣情况下，该提案将解决稳定性问题，即系统是否将保持接近其初始状态或最终将发展一些奇点。Stefanov建议使用各种技术，包括频率分析和规范理论方法来研究这些问题。该项目的另一个共同目标是使这些方法更容易适用于其他问题。波动方程模拟了不同种类的波在物质中的传播。在振动系统和半导体的研究中出现了保守型非线性模型。非线性薛定谔方程描述了激光束在折射率对波幅敏感的介质中的传播。更好地从理论上理解这些模型的时间演化，将使人们能够更好地预测这些系统的物理行为。例如，如果上述类型的物理系统相对于小扰动是稳定的（刚性的），则可以承受系统中存在一定数量的噪声/杂质。对控制这些过程的方程进行分析研究，将有助于更好地理解相应的物理现象。

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.