194 research outputs found
Conformal scattering theory for the linearized gravity fields on Schwarzschild spacetime
We provide in this paper a first step to obtain the conformal scattering
theory for the linearized gravity fields on the Schwarzschild spacetime by
using the conformal geometric approach. We will show that the existing decay
results for the solutions of the Regge-Wheeler and Zerilli equations obtained
recently by L. Anderson, P. Blue and J. Wang \cite{ABlu} is sufficient to
obtain the conformal scattering.Comment: 20 pages, 3 firgure
Cauchy and Goursat problems for the generalized spin zero rest-mass fields on Minkowski spacetime
In this paper, we study the Cauchy and Goursat problems of the spin-
zero rest-mass equations on Minkowski spacetime by using the conformal
geometric method. In our strategy, we prove the wellposedness of the Cauchy
problem in Einstein's cylinder. Then we establish pointwise decays of the
fields and prove the energy equalities of the conformal fields between the null
conformal boundaries \scri^\pm and the hypersurface . Finally, we prove the wellposedness of the Goursat problem in the
partial conformal compactification by using the energy equalities and the
generalisation of H\"ormander's result.Comment: 42 pages, 3 figure
Conformal scattering theory for a tensorial Fackerell-Ipser equation on the Schwarzschild spacetime
In this paper, we prove that the existence of the energy and pointwise decays
for the fields satisfying the tensorial Frackerell-Ipser equations (which are
obtained from the Maxwell and spin Teukolsky equations) on the
Schwarzschild spacetime is sufficient to obtain a conformal scattering theory.
This work is the continuation of the recent work \cite{Pha2020} on the
conformal scattering theory for the Regge-Wheeler and Zerilli equations arising
from the linearized gravity fields and the spin Teukolsky equations.Comment: 27 pages, 3 figures. arXiv admin note: text overlap with
arXiv:2005.1204
Conformal scattering theory for the Dirac field on Kerr spacetime
We investigate to construct a conformal scattering theory of the spin-
massless Dirac equation on the Kerr spacetime by using the conformal geometric
method and under an assumption on the pointwise decay of the Dirac field. In
particular, our construction is valid in the exteriors of Schwarzschild and
very slowly Kerr black hole spacetimes, where the pointwise decay was
established.Comment: 39 pages, 2 figures. arXiv admin note: text overlap with
arXiv:2106.0405
Peeling of Dirac fields on Kerr spacetimes
In a recent paper with J.-P. Nicolas [J.-P. Nicolas and P.T. Xuan, Annales
Henri Poincare 2019], we studied the peeling for scalar fields on Kerr metrics.
The present work extends these results to Dirac fields on the same geometrical
background. We follow the approach initiated by L.J. Mason and J.-P. Nicolas
[L. Mason and J.-P. Nicolas, J.Inst.Math.Jussieu 2009; L. Mason and J.-P.
Nicolas, J.Geom.Phys 2012] on the Schwarzschild spacetime and extended to Kerr
metrics for scalar fields. The method combines the Penrose conformal
compactification and geometric energy estimates in order to work out a
definition of the peeling at all orders in terms of Sobolev regularity near
, instead of regularity at , then
provides the optimal spaces of initial data such that the associated solution
satisfies the peeling at a given order. The results confirm that the analogous
decay and regularity assumptions on initial data in Minkowski and in Kerr
produce the same regularity across null infinity. Our results are local near
spacelike infinity and are valid for all values of the angular momentum of the
spacetime, including for fast Kerr metrics.Comment: 29 page
On periodic solution for the Boussinesq system on real hyperbolic Manifolds
In this work we study the existence and uniqueness of the periodic mild
solutions of the Boussinesq system on the real hyperbolic manifold
(). We will consider Ebin-Marsden's
Laplace operator associated with the corresponding linear system. Our method is
based on the dispertive and smoothing estimates of the semigroup generated by
Ebin-Marsden's Laplace operator. First, we prove the existence and the
uniqueness of the bounded periodic mild solution for the linear system. Next,
using the fixed point arguments, we can pass from the linear system to the
semilinear system to establish the existence of the periodic mild solution.
Finally, we prove the unconditional uniqueness of large periodic mild solutions
for the Boussinesq system on the framework of hyperbolic spaces.Comment: 23 page
Existence and Decay of Solutions of a Nonlinear Viscoelastic Problem with a Mixed Nonhomogeneous Condition
We study the initial-boundary value problem for a nonlinear wave equation
given by u_{tt}-u_{xx}+\int_{0}^{t}k(t-s)u_{xx}(s)ds+ u_{t}^{q-2}u_{t}=f(x,t,u)
, 0 < x < 1, 0 < t < T, u_{x}(0,t)=u(0,t), u_{x}(1,t)+\eta u(1,t)=g(t),
u(x,0)=\^u_{0}(x), u_{t}(x,0)={\^u}_{1}(x), where \eta \geq 0, q\geq 2 are
given constants {\^u}_{0}, {\^u}_{1}, g, k, f are given functions. In part I
under a certain local Lipschitzian condition on f, a global existence and
uniqueness theorem is proved. The proof is based on the paper [10] associated
to a contraction mapping theorem and standard arguments of density. In Part} 2,
under more restrictive conditions it is proved that the solution u(t) and its
derivative u_{x}(t) decay exponentially to 0 as t tends to infinity.Comment: 26 page
Stability for the Boussinesq system on real hyperbolic Manifolds and application
In this paper we study the global existence and stability of mild solution
for the Boussinesq system on the real hyperbolic manifold
(). We will consider a couple of
Ebin-Marsden's Laplace and Laplace-Beltrami operators associated with the
corresponding linear system which provides a vectorial heat semigoup. First, we
prove the existence and the uniqueness of the bounded mild solution for the
linear system by using certain dispersive and smoothing estimates of the
vectorial heat semigroup. Next, using the fixed point arguments, we can pass
from the linear system to the semilinear system to establish the existence of
the bounded mild solution. We will prove the exponential stability of such
solution by using the cone inequality. Finally, we give an application of
stability to the existence of periodic mild solution for the Boussinesq system.Comment: 23 pages. arXiv admin note: substantial text overlap with
arXiv:2209.0780
On asymptotically almost periodic solutions to the Navier-Stokes equations in hyperbolic manifolds
In this paper we extend a recent work \cite{HuyXuan2020} to study the forward
asymptotically almost periodic (AAP-) mild solution of Navier-Stokes equation
on the real hyperbolic manifold with dimension . Using the dispertive and smoothing estimates for Stokes equation
\cite{Pi} we invoke the Massera-type principle to prove the existence and
uniqueness of the AAP- mild solution for the Stokes equation in
space with . We then establish the existence and
uniqueness of the small AAP- mild solutions of the Navier-Stokes equation by
using the fixed point argument. The asymptotic behaviour (exponential decay and
stability) of these small solutions are also related. Our results extend also
\cite{FaTa2013} for the forward asymptotic mild solution of the Navier-Stokes
equation on the curved background.Comment: 21 page
Well-posedness and scattering for wave equations on hyperbolic spaces with singular data
We consider the wave and Klein-Gordon equations on the real hyperbolic space
() in a framework based on weak- spaces.
First, we establish dispersive estimates on Lorentz spaces in the context of
. Then, employing those estimates, we prove global
well-posedness of solutions and an exponential asymptotic stability property.
Moreover, we develop a scattering theory in such singular framework.Comment: 15 page
- …