Partial Differential Equations and Applications
This book contains (updated and expanded) versions of lectures delivered at the school held in Lanzhou (China) in 2004. There are seven contributions on various topics on partial differential equations. B. Grébert describes properties of solutions of Hamiltonian perturbations of integrable systems (the main tool being the Birkhoff normal form and its dynamical consequences - the Birkhoff approach is compared with the Kolmogorov-Arnold-Moser approach). In a long paper, F. Hélein explains many properties of integrable systems on several important examples (sinh-Gordon, Toda, KdV, harmonic maps and ASD Yang-Mills fields). D. Iftimie discusses recent results on large time behaviour of the Euler equations for a perfect incompressible fluid in the plane or half-plane. The role of coherent states in the study of Schrödinger type equations, estimates for their asymptotic solutions and several applications are described in the paper by D. Robert. A short note by W.-M. Wang contains a proof of stability of the bound states for time quasi-periodic perturbations of the quantum harmonic oscillator. A new proof for microlocal resolvent estimates for semi-classical Schrödinger operators with a potential from a special class is presented in the paper by X. P. Wang. Basic methods for a study of standard semi-linear elliptic equations are presented in the last paper by D. Ye.