Mathematical Sciences: Harmonic Analysis Method in Certain Evolution Equations

摘要 杨9532033 建议将谐波分析方法发展到解决 一些非线性发展方程，包括非线性 薛定谔方程，非线性波动方程，修正的 Korteweg-de弗里斯方程、Zakharov方程组和一个四阶 共形几何中出现的椭圆方程。分析中 相关的积分算子，调和分析估计将被用来分析振荡核。已知的结果主体 关于这些方程，主要是从整体上处理方程 空间 需要新的想法和技术来处理这个问题 在有边界的域上。四阶方程需要理解高阶导数的Sobolev不等式。 特别是，寻找类似的正质量定理是必要的。 上述演化方程代表了材料科学和流体力学中的各种模型。比如说 当电子与等离子体发生碰撞时， 温度远大于离子温度， 电子等离子体波（朗缪尔波）和离子声波。下 适当的条件（波的能量密度小， 粒子热能密度和特征尺度大 与德拜长度相比），扎哈罗夫的模型是一个有用的描述 这些集体模型的非线性耦合。Schrodinger 方程可以被理解为Zakharov模型的极限情况，并且可以 也可以用来描述在某些制度的朗缪尔波在一个 等离子体，即，传播包络的时间演化 电场我们的主要项目是分析解决方案的 利用调和分析的工具，例如，我们希望 了解奇点在特定时间的发展， 在薛定谔模型中，代表了朗缪尔波的坍缩。 这一典型问题在模型整体上得到了理解 物理空间我们工作的一部分是研究 模型被限制在物理空间中的有界区域中。的 我们正在做的另一个项目是研究一个非线性几何问题。在这里，我们寻找一个类似于 引力系统的总质量， 对方程的可解性的作用。

Abstract Yang 9532033 The proposed will develop harmonic analysis methods to solve a number of nonlinear evolution equations including the nonlinear Schrodinger equation, the nonlinear wave equation, the modified Korteweg-de Vries equation, the Zakharov systems and a fourth order elliptic equation arising in conformal geometry. In the analysis of the relevant integral operators, harmonic analysis estimates will be used to analyze the oscillatory kernels. The known body of results concerning these equation deal mainly with the equation in the whole space. New ideas and techniques will be needed to handle the problem on domains with boundary. The fourth order equation will require an understanding of the Sobolev inequalities for higher order derivatives. In particular, a search for analogue of the positive mass theorem is required. All of the evolution equations mentioned above represent various models in materials science and fluid mechanics. For example, the collective response of a nearly collisionless plasma when the electron temperature is much greater than the ion temperature is dominated by electron plasma waves (Langmuir waves) and ion acoustic waves. Under appropriate conditions (energy density of the waves small compared to particle thermal energy densities, and characteristic scales large compared to a Debye length), Zakharov's model is a useful description of the nonlinear coupling of these collective models. Schrodinger equation can be understood as a limit case of Zakharov's model and can also be used to describe in certain regimes the Langmuir waves in a plasma, i.e., the time evolution of the envelope of the propagating electric field. Our main project is to analyze the solutions of the equations by using the tools of harmonic analysis. For example, we want to understand the development of singularity at certain time, which represents, in the Schrodinger models, how the Langmuir waves collapse. This typical proble m is understood when the models are on the whole physical space. Part of our work is to study the situation when the models are confined in a bounded region in the physical space. The other project we are working on is to study a non-linear problem in geometry. Here we are looking for a conserved quantity analogous to the total mass of a gravitational systems which will play an important role to the solvability of the equation.