Generator functions and their applications

In [Grenier-Nguyen], we introduced so called {\em generators} functions to precisely follow the regularity of analytic solutions of Navier Stokes equations. In this short note, we give a presentation of these generator functions and use them to give existence results of analytic solutions to some classical equations, namely to hyperbolic equations, to incompressible Euler equations, and to hydrostatic Euler and Vlasov models. The use of these generator functions appear to be an alternative way to the use of the classical abstract Cauchy-Kovalevskaya theorem [Asano,Caflisch,Nirenberg,Safonov].