Paper detail

Unconditional well-posedness for wave maps

We prove uniqueness of solutions to the wave map equation in the natural class, namely $ (u, \partial_t u) \in C([0,T); \dot{H}^{d/2})\times C^1([0,T); \dot{H}^{d/2-1})$ in dimensions $d\geq 4$. This is achieved through estimating the difference of two solutions at a lower regularity level. In order to reduce to the Coulomb gauge, one has to localize the gauge change in suitable cones as well as estimate the difference between the frames and connections associated to each solutions and take advantage of the assumption that the target manifold has bounded curvature.