These notes originate from a graduate course given at the University of Pisa during the <br />spring semester 2007. They were completed while the author was visiting the <br />Centro di Ricerca Matematica Ennio De Giorgi<br />in february 2008. <br />The main objective is to present at the level of beginners an introduction <br />to several modern tools of micro-local analysis which are useful for the mathematical study <br />of nonlinear partial differential equations. The guideline is to show how one can use the para-differential <br />calculus to prove energy estimates using para-differential symmetrizers, or to decouple<br />and reduce systems to equations. In these notes, we have concentrated <br />the applications on the well posed-ness of the Cauchy problem for nonlinear PDE's. <br />These notes are divided in three parts. Part I is an introduction to <br />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 <br />calculus which was introduced by Jean-Michel Bony \cite{Bony} in 1979. Part III is devoted to two applications. <br />First we study quasi-linear hyperbolic systems.<br />The second application concerns the local in time well posedness of the <br />Cauchy problem for systems of Schödinger equations, <br />coupled though quasilinear interactions.< Réduire