Presents a comprehensive exposition of the para-differential calculus for beginners
Provides a continuous exposition of the symmetrizer method from the constant coefficient equation to the variable coefficient case
Details the application to systems of Schrödinger equations
The main aim is to present at the level of beginners several modern tools of micro-local analysis which are useful for the mathematical study of nonlinear partial differential equations. The core of these notes is devoted to a presentation of the para-differential techniques, which combine a linearization procedure for nonlinear equations, and a symbolic calculus which mimics or extends the classical calculus of Fourier multipliers. These methods apply to many problems in nonlinear PDE’s such as elliptic equations, propagation of singularities, boundary value problems, shocks or boundary layers. However, in these introductory notes, we have chosen to illustrate the theory on two selected and relatively simple examples, which allow becoming familiar with the techniques. They concern the well posed-ness of the Cauchy problem for systems of nonlinear PDE's, firstly hyperbolic systems and secondly coupled systems of Schrödinger equations which arise in various models of wave propagation.
I. Introduction to Systems - 1. Notations and Examples - 2. Constant Coefficient Systems. Fourier Synthesis - 3. The Method of Symmetrizers.- II. The Para-Differential Calculus - 1. Pseudo-Differential Operators 2. Para-Differential Operators - 3. Symbolic Calculus.-III. Applications - 1. Nonlinear Hyperbolic systems - 2. Systems of Schrödinger Equations.