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 Schödinger equations,
coupled though quasilinear interactions. |
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