The Hyperbolic Yang–Mills Equation for Connections in an Arbitrary Topological Class

This is the third part of a four-paper sequence, which establishes the Threshold Conjecture and the Soliton-Bubbling versus Scattering Dichotomy for the energy critical hyperbolic Yang–Mills equation in the (4 + 1)-dimensional Minkowski space-time. This paper provides basic tools for considering the dynamics of the hyperbolic Yang–Mills equation in an arbitrary topological class at an optimal regularity. We generalize the standard notion of a topological class of connections on Rd, defined via a pullback to the one-point compactification Sd= Rd∪ { ∞} , to rough connections with curvature in the critical space Ld2(Rd). Moreover, we provide excision and extension techniques for the Yang–Mills constraint (or Gauss) equation, which allow us to efficiently localize Yang–Mills initial data sets. Combined with the results in the previous paper (Oh and Tataru in The hyperbolic Yang–Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions, 2017. arXiv:1709.09332), we obtain local well-posedness of the hyperbolic Yang–Mills equation on R1+d(d≥ 4) in an arbitrary topological class at optimal regularity in the temporal gauge (where finite speed of propagation holds). In addition, in the energy subcritical case d = 3, our techniques provide an alternative proof of the classical finite energy global well-posedness theorem of Klainerman–Machedon (Ann. Math. (2) 142(1):39–119, 1995. https://doi.org/10.2307/2118611), while also removing the smallness assumption in the temporal-gauge local well-posedness theorem of Tao (J. Differ. Equ. 189(2):366–382, 2003. https://doi.org/10.1016/S0022-0396(02)00177-8). Although this paper is a part of a larger sequence, the materials presented in this paper may be of independent and general interest. For this reason, we have organized the paper so that it may be read separately from the sequence

Local energy decay for scalar fields on time dependent non-trapping backgrounds

We consider local energy decay estimates for solutions to scalar wave equations on nontrapping asymptotically flat space-times. Our goals are two-fold. First we consider the stationary case, where we can provide a full spectral characterization of local energy decay bounds; this characterization simplifies in the stationary symmetric case. Then we consider the almost stationary, almost symmetric case. There we establish two main results: The first is a “two point” local energy decay estimate which is valid for a general class of (non-symmetric) almost stationary wave equations which satisfy a certain nonresonance property at zero frequency. The second result, which also requires the almost symmetry condition, is to establish an exponential trichotomy in the energy space via finite dimensional time dependent stable and unstable sub-spaces, with an infinite dimensional complement on which solutions disperse via the usual local energy decay estimate

Renormalization and blow up for charge one equivariant critical wave maps

We prove the existence of equivariant finite time blow up solutions for the wave map problem from 2+1 dimensions into the 2-sphere. These solutions are the sum of a dynamically rescaled ground-state harmonic map plus a radiation term. The local energy of the latter tends to zero as time approaches blow up time. This is accomplished by first "renormalizing" the rescaled ground state harmonic map profile by solving an elliptic equation, followed by a perturbative analysis

Global Schr\"{o}dinger maps

We consider the Schr\"{o}dinger map initial-value problem in dimension two or greater. We prove that the Schr\"{o}dinger map initial-value problem admits a unique global smooth solution, provided that the initial data is smooth and small in the critical Sobolev space. We prove also that the solution operator extends continuously to the critical Sobolev space.Comment: 60 page

Strichartz estimates on Schwarzschild black hole backgrounds

We study dispersive properties for the wave equation in the Schwarzschild space-time. The first result we obtain is a local energy estimate. This is then used, following the spirit of earlier work of Metcalfe-Tataru, in order to establish global-in-time Strichartz estimates. A considerable part of the paper is devoted to a precise analysis of solutions near the trapping region, namely the photon sphere.Comment: 44 pages; typos fixed, minor modifications in several place

Explaining the Better Prognosis of ScreeningExposed Breast Cancers: Influence of Tumor Characteristics and Treatment

This study was funded by a grant from the UK Department of Health (no. 106/0001). The grant was awarded to Prof Stephen W Duffy

Energy dispersed large data wave maps in 2+1 dimensions

In this article we consider large data Wave-Maps from $\mathbb{R}^{2+1}$ into a compact Riemannian manifold $(\mathcal{M},g)$, and we prove that regularity and dispersive bounds persist as long as a certain type of bulk (non-dispersive) concentration is absent. In a companion article we use these results in order to establish a full regularity theory for large data Wave-Maps.Comment: 89 page