24 research outputs found
Small data solutions of the Vlasov-Poisson system and the vector field method
The aim of this article is to demonstrate how the vector field method of
Klainerman can be adapted to the study of transport equations. After an
illustration of the method for the free transport operator, we apply the vector
field method to the Vlasov-Poisson system in dimension 3 or greater. The main
results are optimal decay estimates and the propagation of global bounds for
commuted fields associated with the conservation laws of the free transport
operators, under some smallness assumption. Similar decay estimates had been
obtained previously by Hwang, Rendall and Vel\'azquez using the method of
characteristics, but the results presented here are the first to contain the
global bounds for commuted fields and the optimal spatial decay estimates. In
dimension 4 or greater, it suffices to use the standard vector fields commuting
with the free transport operator while in dimension 3, the rate of decay is
such that these vector fields would generate a logarithmic loss. Instead, we
construct modified vector fields where the modification depends on the solution
itself. The methods of this paper, being based on commutation vector fields and
conservation laws, are applicable in principle to a wide range of systems,
including the Einstein-Vlasov and the Vlasov-Nordstr\"om system
Strong cosmic censorship for T^2-symmetric spacetimes with cosmological constant and matter
We address the issue of strong cosmic censorship for T^2-symmetric spacetimes
with positive cosmological constant. In the case of collisionless matter, we
complete the proof of the C^2 formulation of the conjecture for this class of
spacetimes. In the vacuum case, we prove that the conjecture holds for the
special cases where the area element of the group orbits does not vanish on the
past boundary of the maximal Cauchy development.Comment: 29 pages, 6 figures, v2 is the version published in Annales Henri
Poincar
A vector field method for relativistic transport equations with applications
We adapt the vector field method of Klainerman to the study of relativistic
transport equations. First, we prove robust decay estimates for velocity
averages of solutions to the relativistic massive and massless transport
equations, without any compact support requirements (in or ) for the
distribution functions. In the second part of this article, we apply our method
to the study of the massive and massless Vlasov-Nordstr\"om systems. In the
massive case, we prove global existence and (almost) optimal decay estimates
for solutions in dimensions under some smallness assumptions. In the
massless case, the system decouples and we prove optimal decay estimates for
the solutions in dimensions for arbitrarily large data, and in
dimension under some smallness assumptions, exploiting a certain form of
the null condition satisfied by the equations. The -dimensional massive case
requires an extension of our method and will be treated in future work.Comment: 72 pages, 3 figure
Global geometry of T2 symmetric spacetimes with weak regularity
We define the class of weakly regular spacetimes with T2 symmetry, and
investigate their global geometry structure. We formulate the initial value
problem for the Einstein vacuum equations with weak regularity, and establish
the existence of a global foliation by the level sets of the area R of the
orbits of symmetry, so that each leaf can be regarded as an initial
hypersurface. Except for the flat Kasner spacetimes which are known explicitly,
R takes all positive values. Our weak regularity assumptions only require that
the gradient of R is continuous while the metric coefficients belong to the
Sobolev space H1 (or have even less regularity).Comment: 5 page
The Stability of the Minkowski space for the Einstein-Vlasov system
We prove the global stability of the Minkowski space viewed as the trivial
solution of the Einstein-Vlasov system. To estimate the Vlasov field, we use
the vector field and modified vector field techniques developed in [FJS15;
FJS17]. In particular, the initial support in the velocity variable does not
need to be compact. To control the effect of the large velocities, we identify
and exploit several structural properties of the Vlasov equation to prove that
the worst non-linear terms in the Vlasov equation either enjoy a form of the
null condition or can be controlled using the wave coordinate gauge. The basic
propagation estimates for the Vlasov field are then obtained using only weak
interior decay for the metric components. Since some of the error terms are not
time-integrable, several hierarchies in the commuted equations are exploited to
close the top order estimates. For the Einstein equations, we use wave
coordinates and the main new difficulty arises from the commutation of the
energy-momentum tensor, which needs to be rewritten using the modified vector
fields.Comment: 139 page
Quasimodes and a Lower Bound on the Uniform Energy Decay Rate for Kerr-AdS Spacetimes
We construct quasimodes for the Klein-Gordon equation on the black hole
exterior of Kerr-Anti-de Sitter (Kerr-AdS) spacetimes. Such quasi-modes are
associated with time-periodic approximate solutions of the Klein Gordon
equation and provide natural candidates to probe the decay of solutions on
these backgrounds. They are constructed as the solutions of a semi-classical
non-linear eigenvalue problem arising after separation of variables, with the
(inverse of the) angular momentum playing the role of the semi-classical
parameter. Our construction results in exponentially small errors in the
semi-classical parameter. This implies that general solutions to the Klein
Gordon equation on Kerr-AdS cannot decay faster than logarithmically. The
latter result completes previous work by the authors, where a logarithmic decay
rate was established as an upper bound