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-$n/2$ 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 $\Sigma_0=\left\{ t=0 \right\}$. 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 $\pm 1$ 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 $\pm 2$ 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-$1/2$ 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 $\mathscr{I}$, instead of ${\mathcal C}^k$ regularity at $\mathscr{I}$, 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 $\mathbb{H}^d(\mathbb{R})$ ($d \geqslant 2$). 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 $\mathbb{H}^d(\mathbb{R})$ ($d \geqslant 2$). 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 $\mathbb{H}^d(\mathbb{R})$ with dimension $d \geq 2$. 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 $L^p(\Gamma(TM)))$ space with $p>d$. 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 $\mathbb{H}^{n}$ ($n \geq2$) in a framework based on weak-$L^{p}$ spaces. First, we establish dispersive estimates on Lorentz spaces in the context of $\mathbb{H}^{n}$. 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