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The nonlinear Hartree equation describes the macroscopic dynamics of initially factorized N-boson states, in the limit of large N. In this paper we provide estimates on the rate of convergence of the… (More)

We consider solutions to the linear wave equation on a (maximally extended) Schwarzschild spacetime, assuming only that the solution decays suitably at spatial infinity on a complete Cauchy… (More)

These lecture notes, based on a course given at the Zürich Clay Summer School (June 23–July 18 2008), review our current mathematical understanding of the global behaviour of waves on black hole… (More)

We present a new general method for proving global decay of energy through a suitable spacetime foliation, as well as pointwise decay, starting from an integrated local energy decay estimate. The… (More)

In this paper we establish dispersive estimates for solutions to the linear Schrödinger equation in three dimensions 0.1$$\frac{1}{i}\partial_t \psi - \triangle \psi + V\psi = 0,\qquad \psi(s)=f$$… (More)

We give a new proof of the global stability of Minkowski space originally established in the vacuum case by Christodoulou and Klainerman. The new approach, which relies on the classical harmonic… (More)

A well-known open problem in general relativity, dating back to 1972, has been to prove Price’s law for an appropriate model of gravitational collapse. This law postulates inverse-power decay rates… (More)

We prove global stability of Minkowski space for the Einstein vacuum equations in harmonic (wave) coordinate gauge for the set of restricted data coinciding with the Schwarzschild solution in the… (More)

We prove here essentially sharp L q (L r) linear and bilinear estimates for the wave equations on Minkowski space where we assume the initial data possesses additional regularity with respect to… (More)

Abstract.We show that in dimensions n ≥ 6 one has global regularity for the Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical Sobolev norm of the initial data is… (More)