A Class Of Large Global Solutions For The Wave-Map Equation

In this paper we consider the equation for equivariant wave maps from R3+1 to S-3 and we prove global in forward time existence of certain C-infinity-smooth solutions which have infinite critical Sobolev norm (H) overdot(3/4) (R-3) x (H) overdot(1/2) (R-3). Our construction provides solutions which can moreover satisfy the additional size condition parallel to u(0, .)parallel to L-infinity(vertical bar-vertical bar >= 1) > M for arbitrarily chosen M > 0. These solutions are also stable under suitable perturbations. Our method, strongly inspired by work of Krieger and Schlag, is based on a perturbative approach around suitably constructed approximate self similar solutions.