Galerkin averaging method and Poincaré normal form for some quasilinear PDEs

GALERKIN AVERAGING METHOD AND POINCARÉ NORMAL FORM FOR SOME QUASILINEAR PDEs

We use the Galerkin averaging method to construct a coordinate transformation putting a nonlinear PDE in Poincaré normal form up to a remainder of arbitrary order. The abstract theorem we obtain applies to quite general quasilinear equations in one or more space dimensions. Applications to nonlinear wave and heat equations are given. Dynamical consequences are obtained. Mathematics Subject Clas...

BIRKHOFF NORMAL FORM FOR SOME NONLINEAR PDEs

AVERAGING THEOREM FOR QUASILINEAR HAMILTONIAN PDEs IN ARBITRARY SPACE DIMENSIONS

We study the dynamics of quasilinear Hamiltonian wave equations with Dirichlet boundary conditions in an n–dimensional parallepided. We prove an averaging theorem according to which the solution corresponding to an arbitrary small amplitude smooth initial datum remains arbitrarily close to a finite dimensional torus up to very long times. We expect the result to be valid for a very general clas...

A dissipative Galerkin method applied to some quasilinear hyperbolic equations

— À nonstandard continuous-in-time Galerkin method, based on piecewise polynomial spaces, is applied io the periodic initial value problem for the équation ut = a(x, ty u)ux + ƒ(*, ty «). Under the condition that a(x, t, u) > «o > 0 for the solution, optimal order L error estimâtes are derived.

Birkhoff Normal Form and Hamiltonian PDEs

These notes are based on lectures held at the Lanzhou university (China) during a CIMPA summer school in july 2004 but benefit from recent devellopements. Our aim is to explain some perturbations technics that allow to study the long time behaviour of the solutions of Hamiltonian perturbations of integrable systems. We are in particular interested with stability results. Our approach is centere...