Besov spaces and self-similar solutions for the wave-map equation

Self-similar solutions play a crucial role in the blow-up theory for the wave-map equation; they correspond to self-similar data at the time of the blow-up. However, solutions to this equation are generally considered for data in the standard finite energy spaces (in dimension d) H^d/2 × H^d/2-1. We build up in this article solutions of the covariant wave-map equation for data which are small and of infinite energy, or large and self-similar. This provides us with a general framework which includes in particular the blowing up solutions of Shatah [14] and Bizon [3]. As an application, we describe more precisely the blow-up phenomenon.