Harmonic Analysis and Nonlinear Dispersive Equations

调和分析和非线性色散方程建议研究摘要这个项目集中于使用傅立叶分析的方法来分析数学物理中出现的某些非线性偏微分方程式。我们主要感兴趣的是色散方程，其中薛定谔算子或波算符是主要的阶项。特别是研究Maxwell-Schroedinger、Maxwell-Klein-Gordon和Maxwell-Dirac系统。我们只假定初始数据是“粗略的”。这在物理上更现实，需要更谨慎地使用基本的守恒定律。我们希望得到关于这些方程解的适定性、正则性和渐近性的结果。我们期待傅立叶分析、谱理论和规范理论方法的新工具将被开发和使用来证明这些结果。目前的提议将产生新的数学工具来研究非线性色散方程，这些方程是重要过程的数学模型，如材料的磁化、光在光学介质中的传播以及量子力学的相关现象。其中一些模型出现在量子力学系统和非线性光学的研究中，而另一些模型则纯粹是几何起源。更好地描述这些方程的解的性质，特别是它们在时间和空间上的渐近行为，将极大地提高我们对这些非线性场论性质的理解，并有助于发展使用它们的技术。

Harmonic Analysis and Nonlinear Dispersive EquationsAbstract of Proposed Research Atanas G StefanovThis project centers on the use of methods from Fourier analysis to analyze certain nonlinear partial differential equations arising in mathematical physics. Our primary interest is on dispersive equations with either Schroedinger or wave operators as the leading order terms. In particular the Maxwell-Schroedinger, Maxwell-Klein-Gordon and the Maxwell-Dirac systems will be studied. Only ``rough'' initial data will be assumed. This is more realistic physically and requires more careful use of the underlying conservation laws. We want to obtain results about the well-posedness, regularity and asymptotic behavior of the solutions of these equations.. We expect that new tools from Fourier analysis, spectral theory and gauge theoretic methods will be developed and used to prove these results. The current proposal will produce new mathematical tools to study nonlinear dispersive equations, which are mathematical models for important processes, such as magnetization of materials, propagation of light in optical medium and related phenomena of quantum mechanics. Some of these models arise in the study of quantum mechanical systems and nonlinear optics, while others have purely geometric origins. Better mathematical description of the properties of the solutions of these equations, especially their asymptotic behavior in time and space, will greatly improve our understanding of the properties of these nonlinear field theories and help in the development of technologies that use them.