RENORMALIZATION AND BLOW-UP FOR WAVE MAPS FROM S-2 x R TO S-2

We construct a one parameter family of finite time blow-ups to the co-rotational wave maps problem from S-2 x R to S-2, parameterized by nu is an element of (1/2, 1]. The longitudinal function u(t, alpha) which is the main object of study will be obtained as a perturbation of a rescaled harmonic map of rotation index one from R-2 to S-2. The domain of this harmonic map is identified with a neighborhood of the north pole in the domain S-2 via the exponential coordinates (alpha, theta). In these coordinates u(t, alpha) = Q(lambda(t)alpha) + R(t, alpha), where Q(r) = 2arctan r is the standard co-rotational harmonic map to the sphere, lambda(t) = t(-1-nu), and R(t, alpha) is the error with local energy going to zero as t -> 0. Blow-up will occur at (t, alpha) = (0, 0) due to energy concentration, and up to this point the solution will have regularity H1+nu-.