318 research outputs found
On the well-posedness and scattering for the Gross-Pitaevskii hierarchy via quantum de Finetti
We prove the existence of scattering states for the defocusing cubic
Gross-Pitaevskii (GP) hierarchy in . Moreover, we show that an
energy growth condition commonly used in the well-posedness theory of the GP
hierarchy is, in a specific sense, necessary. In fact, we prove that without
the latter, there exist initial data for the focusing cubic GP hierarchy for
which instantaneous blowup occurs.Comment: AMS Latex, 19 page
Theorems on existence and global dynamics for the Einstein equations
This article is a guide to theorems on existence and global dynamics of
solutions of the Einstein equations. It draws attention to open questions in
the field. The local-in-time Cauchy problem, which is relatively well
understood, is surveyed. Global results for solutions with various types of
symmetry are discussed. A selection of results from Newtonian theory and
special relativity that offer useful comparisons is presented. Treatments of
global results in the case of small data and results on constructing spacetimes
with prescribed singularity structure or late-time asymptotics are given. A
conjectural picture of the asymptotic behaviour of general cosmological
solutions of the Einstein equations is built up. Some miscellaneous topics
connected with the main theme are collected in a separate section.Comment: Submitted to Living Reviews in Relativity, major update of Living
Rev. Rel. 5 (2002)
Global existence for the spherically symmetric Einstein-Vlasov system with outgoing matter
We prove a new global existence result for the asymptotically flat,
spherically symmetric Einstein-Vlasov system which describes in the framework
of general relativity an ensemble of particles which interact by gravity. The
data are such that initially all the particles are moving radially outward and
that this property can be bootstrapped. The resulting non-vacuum spacetime is
future geodesically complete.Comment: 16 page
Extremal black hole formation as a critical phenomenon
In this paper, we prove that extremal black holes arise on the threshold of
gravitational collapse. More precisely, we construct smooth one-parameter
families of smooth, spherically symmetric solutions to the
Einstein-Maxwell-Vlasov system which interpolate between dispersion and
collapse and for which the critical solution is an extremal black hole.
Physically, these solutions can be understood as beams of gravitationally
self-interacting collisionless charged particles fired into Minkowski space
from past infinity. Depending on the precise value of the parameter, we show
that the Vlasov matter either disperses due to the combined effects of angular
momentum and electromagnetic repulsion, or undergoes gravitational collapse. At
the critical value of the parameter, an extremal Reissner-Nordstr\"om black
hole is formed. No naked singularities occur as the extremal threshold is
crossed. We call this critical phenomenon extremal critical collapse and the
present work constitutes the first rigorous result on the black hole formation
threshold in general relativity.Comment: 91 pages + references, 16 figure
Formation of trapped surfaces for the spherically symmetric Einstein-Vlasov system
We consider the spherically symmetric, asymptotically flat, non-vacuum
Einstein equations, using as matter model a collisionless gas as described by
the Vlasov equation. We find explicit conditions on the initial data which
guarantee the formation of a trapped surface in the evolution which in
particular implies that weak cosmic censorship holds for these data. We also
analyze the evolution of solutions after a trapped surface has formed and we
show that the event horizon is future complete. Furthermore we find that the
apparent horizon and the event horizon do not coincide. This behavior is
analogous to what is found in certain Vaidya spacetimes. The analysis is
carried out in Eddington-Finkelstein coordinates.Comment: 2
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