Harmonic analysis and applications to geometric PDE's

PI：Atanas G.Stefan ovDMS-0300511摘要：该提案涉及几何学和电磁学理论中出现的几个问题。主要目的是分析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：波动映射问题、薛定谔映射问题和杨-米尔斯场方程。遵循每个物理系统都试图最小化其能量的基本物理原理，形成并研究了这些非线性偏微分方程解，它们作为相应的能量泛函的极小化而产生。这些方程的存在性、唯一性和稳定性问题已经在不同的环境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。斯特凡诺夫将专注于证明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光滑度，以显示解在时间上全局存在。在物理上有趣的小能量数据的情况下(对非常接近其平衡的系统进行建模)，该提议将解决稳定性问题，即系统是否将保持接近其初始状态或最终将发展出一些奇点。斯特凡诺夫建议使用包括频率分析和规范理论方法在内的各种技术来研究这些问题。该项目的一个共同目标是使这些方法更容易适用于其他问题。波动方程模拟不同种类的波在材料中的传播。在对振动系统和半导体的研究中，出现了守恒型的非线性模型。非线性薛定谔方程描述了激光在折射率对波幅敏感的介质中的传输。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此类系统的物理行为。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)，则可以承受系统中一定数量的噪声/杂质。对支配这种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.