Harmonic analysis and applications to geometric PDE's

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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.