We prove that the critical Maxwell-Klein Gordon equation on R 4+1 is globally well-posed for smooth initial data which are small in the energy. This reduces the problem of global regularity for large, smooth initial data to precluding concentration of energy.

We provide a uniform decay estimate for the local energy of general solutions to the inhomogeneous wave equation on a Schwarzchild background. Our estimate implies that such solutions have asymptotic behavioras long as the source term is bounded in the normIn particular this gives scattering at small amplitudes for non-linear scalar fields of the form ✷gφ = λ|φ| p φ for all 2 < p.

We study the phenomena of energy concentration for the critical O(3) sigma model, also known as the wave map flow from R 2+1 Minkowski space into the sphere S 2 . We establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities. The construction and analysis is done in the context of the k-equivariant symmetry reduction, and we restrict to maps with homotopy class k 4. The concentration mechanism we uncover is essentially due to a resonant self-focusing (shrinking) of a corresponding harmonic map. We show that the phenomenon is generic (e.g. in certain Sobolev spaces) in that it persists under small perturbations of initial data, while the resulting blowup is bounded by a log-modified self-similar asymptotic.

Abstract:In this article we prove a Sacks-Uhlenbeck/Struwe type global regularity result for wave-maps : R 2+1 → M into general compact target manifolds M.

Abstract:In this article we consider large data Wave-Maps from R 2+1 into a compact Riemannian manifold (M, g), and we prove that regularity and dispersive bounds persist as long as a certain type of bulk (non-dispersive) concentration is absent. This is a companion to our concurrent article [21], which together with the present work establishes a full regularity theory for large data Wave-Maps.