Mathematical Scienecs: Linear and Nonlinear Waves

9401777将调查两个主要主题。在非线性波动方程的柯西问题中，Ginibre-Soffer-Velo关于临界功率非线性波动方程的结果对于径向情况是完备的，将进一步发展以包含一般数据。它是基于应用新的勒贝格第p次方权界，它允许根据奇异加权范数和部分正则性来控制函数的p范数。在三体色散系统理论中，将使用一类新的伸缩，通过各种单位划分变形为团簇分解。这将允许证明三体色散方程的局部衰变和其他光谱性质。现代物理学，量子力学和相对论，是二十世纪的产物。它是在上个世纪试图解决物质的微观结构和掌握远距离作用力、电磁学和热辐射的概念时坚定地建立起来的。这些发展的数学基础统称为数学物理，从控制粒子动力学的薛定谔算符的详细分析，到试图将四种已知力统一为一种力的统一场论。***

9401777 Soffer Two main topics will be investigated. In the Cauchy problem for the nonlinear wave equation, the results of Ginibre-Soffer- Velo on the critical power nonlinear wave equation which are complete for the radial case will be further developed to include the general data. It is based on applying new Lebesgue pth power bounds which allow the control of the p-norm of a function in terms of singular weighted norms and partial regularity. In the theory of three body dispersive systems a new class of dilations, deformed by various partitions of unity to cluster decompositions will be used. This will allow proofs of local decay and other spectral properties of three body dispersive equations. Modern physics, quantum mechanics and relativity, is a product of the twentieth century. It is founded firmly in the last century's attempt to address the microstructure of matter and to come to grips with the concept of action-at-a distance, electro-magnetism, and heat radiation. The mathematical foundations for these developments collectively called mathematical physics, ranges from detailed analysis of Schroedinger operators, which governs the dynamics of particles, to unified field theory, which attempts to unite the four known forces into a single force. ***