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

Carleman estimates and absence of embedded eigenvalues

Let L be a Schroedinger operator with potential W in L^{(n+1)/2}. We prove that there is no embedded eigenvalue. The main tool is an Lp Carleman type estimate, which builds on delicate dispersive estimates established in a previous paper. The arguments extend to variable coefficient operators with long range potentials and with gradient potentials.Comment: 26 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

Concerning the Wave equation on Asymptotically Euclidean Manifolds

We obtain KSS, Strichartz and certain weighted Strichartz estimate for the wave equation on $(\R^d, \mathfrak{g})$, $d \geq 3$, when metric $\mathfrak{g}$ is non-trapping and approaches the Euclidean metric like $x ^{- \rho}$ with $\rho>0$. Using the KSS estimate, we prove almost global existence for quadratically semilinear wave equations with small initial data for $\rho> 1$ and $d=3$. Also, we establish the Strauss conjecture when the metric is radial with $\rho>0$ for $d= 3$.Comment: Final version. To appear in Journal d'Analyse Mathematiqu

On the 2d Zakharov system with L^2 Schr\"odinger data

We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L^2 x H^{-1/2} x H^{-3/2}. This is the space of optimal regularity in the sense that the data-to-solution map fails to be smooth at the origin for any rougher pair of spaces in the L^2-based Sobolev scale. Moreover, it is a natural space for the Cauchy problem in view of the subsonic limit equation, namely the focusing cubic nonlinear Schroedinger equation. The existence time we obtain depends only upon the corresponding norms of the initial data - a result which is false for the cubic nonlinear Schroedinger equation in dimension two - and it is optimal because Glangetas-Merle's solutions blow up at that time.Comment: 30 pages, 2 figures. Minor revision. Title has been change

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