Please recommend some classical books or articles on the local well-posedness result of compressible Euler equations ! The main aim is that I want to learn some basic methods and techniques about the local existence theory of quasilinear hyperbolic system. Eventually, I can apply the tools I mastered to solve the Euler-Poisson system (Of course, I need to the help of some classical articles). ^-^
By far, the ever best written book on the subject is that of Constantin Dafermos: Hyperbolic Conservation Law in Continuum Physics. Grundlehren der Mathematischen Wissenschaften 325, Springer Verlag.
Edit. Of course, if you are already a bit familiar with the topic, you may have a look to the book by S. Benzoni-Gavage and I: Multi-dimensional hyperbolic partial differential equations. First order systems and applications. Oxford University Press (2007).