Normal form and dynamics of the Kirchhoff equation

We summarize some recent results on the Cauchy problem for the Kirchhoff equation on the d-dimensional torus T^d, with initial data of size epsilon in Sobolev class. While the standard local theory gives an existence time of order epsilon^(-2), a quasilinear normal form allows to give a lower bound on the existence time of the order of epsilon^(−4) for all initial data, improved to epsilon^(−6) for initial data satisfying a suitable nonresonance condition. We also use such a normal form in an ongoing work with F. Giuliani and M. Guardia to prove existence of chaotic-like motions for the Kirchhoff equation.