Para-Differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems
Para-Differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems
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发表时间：
2008-08
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通讯作者：
G. Métivier
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作者：
G. Métivier
这些笔记来自比萨大学2007年春季学期的一门研究生课程。2008年2月，作者在访问埃尼奥·德·乔治数学研究中心时完成了这些研究。主要目的是在初学者的水平上介绍几种对非线性偏微分方程的数学研究有用的微局部分析的现代工具。指导方针是展示如何使用准微分演算来证明使用准微分对称器的能量估计，或解耦并将系统简化为方程。在这些笔记中，我们集中讨论了非线性偏微分方程的柯西问题的适定性。这些音符分为三部分。第一部分是对演化方程的介绍。在给出物理实例的基础上，给出了常系数系统的分析依据。在第二部分中，我们给出了由Jean-Michel Bony \cite{Bony}在1979年引入的准微分学的一个基本的和独立的表示。第三部分主要讨论两种应用。首先我们研究拟线性双曲系统。第二个应用涉及通过拟线性相互作用耦合的薛定谔方程组的柯西问题的局部时间适定性。
These notes originate from a graduate course given at the University of Pisa during the spring semester 2007. They were completed while the author was visiting the Centro di Ricerca Matematica Ennio De Giorgi in february 2008. The main objective is to present at the level of beginners an introduction to several modern tools of micro-local analysis which are useful for the mathematical study of nonlinear partial differential equations. The guideline is to show how one can use the para-differential calculus to prove energy estimates using para-differential symmetrizers, or to decouple and reduce systems to equations. In these notes, we have concentrated the applications on the well posed-ness of the Cauchy problem for nonlinear PDE's. These notes are divided in three parts. Part I is an introduction to evolution equations. After the presentation of physical examples, we give the bases of the analysis of systems with constant coefficients. In Part II, we give an elementary and self-contained presentation of the para-differential calculus which was introduced by Jean-Michel Bony \cite{Bony} in 1979. Part III is devoted to two applications. First we study quasi-linear hyperbolic systems. The second application concerns the local in time well posedness of the Cauchy problem for systems of Schodinger equations, coupled though quasilinear interactions.