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Lectures on black holes and linear waves
These lecture notes, based on a course given at the Zurich Clay Summer School (June 23-July 18, 2008), review our current mathematical understanding of the global behaviour of waves on black hole
The red‐shift effect and radiation decay on black hole spacetimes
We consider solutions to the linear wave equation □gϕ = 0 on a (maximally extended) Schwarzschild spacetime with parameter M > 0, evolving from sufficiently regular initial data prescribed on a
The global stability of Minkowski space-time in harmonic gauge
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
Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems
We exhibit stable finite time blow up regimes for the energy critical co-rotational Wave Map with the S2 target in all homotopy classes and for the critical equivariant SO(4) Yang-Mills problem. We
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics
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
Time decay for solutions of Schrödinger equations with rough and time-dependent potentials
In this paper we establish dispersive estimates for solutions to the linear Schrödinger equation in three dimensions 0.1$$\frac{1}{i}\partial_t \psi - \triangle \psi + V\psi = 0,\qquad \psi(s)=f$$
The linear stability of the Schwarzschild solution to gravitational perturbations
We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a
Global Existence for the Einstein Vacuum Equations in Wave Coordinates
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
A new physical-space approach to decay for the wave equation with applications to black hole spacetimes
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
A proof of Price’s law for the collapse of a self-gravitating scalar field
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