Normal form and dynamics of the Kirchhoff equation

We summarize some recent results on the Cauchy problem for the Kirchhoff equation partial derivative(tt)u - triangle u (1 + integral Td |del u|(2) = 0 on the d-dimensional torus Td, with initial data u(0, x), 8tu(0, x) of size 6 in Sobolev class. While the standard local theory gives an existence time of order 6-2, a quasilinear normal form allows to give a lower bound on the existence time of the order of 6-4 for all initial data, improved to 6-6 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.