Scattering Below Critical Energy for the Radial 4D Yang-Mills Equation and for the 2D Corotational Wave Map System

Given g and f  =  gg′, we consider solutions to the following non linear wave equation : Under suitable assumptions on g, this equation admits non-constant stationary solutions : we denote Q one with least energy. We characterize completely the behavior as time goes to  ±∞ of solutions (u, u _t ) corresponding to data with energy less than or equal to the energy of Q : either it is (Q, 0) up to scaling, or it scatters in the energy space. Our results include the cases of the 2 dimensional corotational wave map system, with target , in the critical energy space, as well as the 4 dimensional, radially symmetric Yang-Mills fields on Minkowski space, in the critical energy space. Centre National de la Recherche Scientifique. Institut des Hautes études Scientifiques. The work of R.C. and F.M. has been supported in part by ANR grant ONDE NONLIN, and the work of C.E.K. has been supported in part by NSF.