Walter CRAIG - Birkhoff normal form for nonlinear wave equations
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Date: 2016-06-03

Many theorems on global existence of small amplitude solutions of nonlinear wave equations  in  ${\mathbb  R}^n$  depend  upon  a  competition  between  the  time decay  of  solutions  and  the  degree  of  the  nonlinearity.  Decay  estimates  are  more effective when inessential nonlinear terms are able to be removed through a well-chosen transformation. Most wave equations that arise in a physical context can be considered  as  Hamiltonian  PDEs,  that  is,  partial  differential  equations  that  can  be formulated as a Hamiltonian system. In this talk, we construct Birkhoff normal forms transformations  for  the  class  of  wave  equations  which  are  Hamiltonian  PDEs  and null forms, giving a new proof via canonical transformations of the global existence theorems  for  null  form  wave  equations of  S.  Klainerman,  J.  Shatah  and  other,  in space dimensions $n \geq 3$.The critical case $n = 2$ is also under consideration. These results are work-in-progress with A. French and C.
-
R. Yang