461 research outputs found
The Strauss conjecture on asymptotically flat space-times
By assuming a certain localized energy estimate, we prove the existence
portion of the Strauss conjecture on asymptotically flat manifolds, possibly
exterior to a compact domain, when the spatial dimension is 3 or 4. In
particular, this result applies to the 3 and 4-dimensional Schwarzschild and
Kerr (with small angular momentum) black hole backgrounds, long range
asymptotically Euclidean spaces, and small time-dependent asymptotically flat
perturbations of Minkowski space-time. We also permit lower order perturbations
of the wave operator. The key estimates are a class of weighted Strichartz
estimates, which are used near infinity where the metrics can be viewed as
small perturbations of the Minkowski metric, and the assumed localized energy
estimate, which is used in the remaining compact set.Comment: Final version, to appear in SIAM Journal on Mathematical Analysis. 17
page
Long time existence for semilinear wave equations on asymptotically flat space-times
We study the long time existence of solutions to nonlinear wave equations
with power-type nonlinearity (of order ) and small data, on a large class of
-dimensional nonstationary asymptotically flat backgrounds, which
include the Schwarzschild and Kerr black hole space-times. Under the assumption
that uniform energy bounds and a weak form of local energy estimates hold
forward in time, we give lower bounds of the lifespan when and is
not bigger than the critical one. The lower bounds for three dimensional
subcritical and four dimensional critical cases are sharp in general. For the
most delicate three dimensional critical case, we obtain the first existence
result up to , for many space-times including the
nontrapping exterior domain, nontrapping asymptotically Euclidean space and
Schwarzschild space-time.Comment: Final version, to appear in Communications in Partial Differential
Equations. 24 page
Global existence for a coupled wave system related to the Strauss conjecture
A coupled system of semilinear wave equations is considered, and a small data
global existence result related to the Strauss conjecture is proved. Previous
results have shown that one of the powers may be reduced below the critical
power for the Strauss conjecture provided the other power sufficiently exceeds
such. The stability of such results under asymptotically flat perturbations of
the space-time where an integrated local energy decay estimate is available is
established.Comment: 12 page
Local decay of waves on asymptotically flat stationary space-times
In this article we study the pointwise decay properties of solutions to the
wave equation on a class of stationary asymptotically flat backgrounds in three
space dimensions. Under the assumption that uniform energy bounds and a weak
form of local energy decay hold forward in time we establish a local
uniform decay rate for linear waves. This work was motivated by open problems
concerning decay rates for linear waves on Schwarzschild and Kerr backgrounds,
where such a decay rate has been conjectured by R. Price. Our results apply to
both of these cases.Comment: 33 pages; minor corrections, updated reference
The Einstein-Vlasov system/Kinetic theory
The main purpose of this article is to provide a guide to theorems on global
properties of solutions to the Einstein-Vlasov system. This system couples
Einstein's equations to a kinetic matter model. Kinetic theory has been an
important field of research during several decades in which the main focus has
been on nonrelativistic and special relativistic physics, {\it i.e.} to model
the dynamics of neutral gases, plasmas, and Newtonian self-gravitating systems.
In 1990, Rendall and Rein initiated a mathematical study of the Einstein-Vlasov
system. Since then many theorems on global properties of solutions to this
system have been established. The Vlasov equation describes matter
phenomenologically and it should be stressed that most of the theorems
presented in this article are not presently known for other such matter models
({\it i.e.} fluid models). This paper gives introductions to kinetic theory in
non-curved spacetimes and then the Einstein-Vlasov system is introduced. We
believe that a good understanding of kinetic theory in non-curved spacetimes is
fundamental to good comprehension of kinetic theory in general relativity.Comment: 40 pages, updated version, to appear in Living Reviews in Relativit
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