| We study wave maps equation in three distinct settings. First, we prove a small data result for wave maps on a curved background. We show global existence and uniqueness for initial data that is small in the critical norm in the case that the background manifold is a small perturbation of the Euclidean space. Next, we establish relaxation of an arbitrary one-equivariant wave map exterior to the unit ball in three space dimensions and to the three-sphere of finite energy and with a Dirichlet condition on the boundary of the ball, to the unique stationary harmonic map in its degree class. This settles a recent conjecture of Bizon, Chmaj, and Maliborski who observed this asymptotic behavior numerically, and can be viewed as a verification of the soliton resolution conjecture for this particular model. The chapters concerning these results are based on joint work with Wilhelm Schlag, and with Carlos Kenig and W. Schlag. In the final two chapters, we consider one-equivariant wave maps from two dimensional Minkowski space to the two-sphere. For wave maps of topological degree zero we prove global existence and scattering for energies below twice the energy of harmonic map, Q, given by stereographic projection. This gives a proof in the equivariant case of a refined version of the threshold conjecture adapted to the degree zero theory where the true threshold is two times the energy of Q. The aforementioned global existence and scattering statement can also be deduced by considering the work of Sterbenz and Tataru in the equivariant setting. For wave maps of topological degree one, we establish a classification of solutions blowing up in finite time with energies less than three times the energy of Q. Under this restriction on the energy, we show that a blow-up solution of degree one decouples as it approaches the blow-up times into the sum of a rescaled Q plus a remainder term of topological degree zero of energy less than twice the energy of Q. This result reveals the universal character of the known blow-up constructions for degree one, one-equivariant wave maps of Krieger, Schlag, and Tataru as well as Raphael and Rodnianski. Lastly, we deduce a classification of all degree one global solutions whose energies are less than three times the energy of the harmonic map Q. In particular, for each global energy solution of topological degree one, we show that the solution asymptotically decouples into a rescaled harmonic map plus a radiation term. Together with the degree one finite time blow-up result, this gives a characterization of all one-equivariant, degree one wave maps with energy up to three times the energy of Q. The last two chapters are based on joint work with Raphael Cote, C. Kenig, and W. Schlag. |
