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

Abstract

Given g and f = gg′, we consider solutions to the following non linear wave equation : $$\left\{ \begin{array}{l}\displaystyle u_{tt} - u_{rr} - \frac 1 r u_r = - \frac{f(u)}{r^2},\\(u, u_t)|_{t =0} = (u_0,u_1). \end{array}\right.$$ 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 $${{\mathbb S}^2}$$ , in the critical energy space, as well as the 4 dimensional, radially symmetric Yang-Mills fields on Minkowski space, in the critical energy space.