Alain
BACHELOT
Professeur émérite
Université de Bordeaux
Alain
BACHELOT
Professeur émérite
Université de Bordeaux
Dernière mise à jour, 21 décembre 2018.
à l'Ecole Normale Supérieure de Rabat-Souissi, Maroc :
à l'Université de Bordeaux :
Conseiller scientifique au CEA (2000-2008).
Articles
dans des revues à comité de
lecture / Articles
in Journal with referee
Conférences publiées avec comité de lecture / Proceedings with referee
http://www.ladocumentationfrancaise.fr/rapports-publics/064000887/index.shtml
A. BACHELOT. Les
mathématiques des trous noirs, in Brèves de maths,
Martin Andler, Liliane Bel, Sylvie Benzoni, Thierry Goudon, Cyril
Imbert, Antoine Rousseau, Editions Nouveau Monde, 2014, en ligne sur
A. BACHELOT. Voir les trous noirs, in Brèves de maths, Martin Andler, Liliane Bel, Sylvie Benzoni, Thierry Goudon, Cyril Imbert, Antoine Rousseau, Editions Nouveau Monde, 2014, en ligne sur
http://www.breves-de-maths.fr/voir-les-trous-noirs/Retour à la page d'accueil. / Back to Home Page
- A. BACHELOT. The Dirac system on the Anti-de Sitter Universe. http://arxiv.org/abs/0706.1315.
- A. BACHELOT. Wave Propagation and Scattering for the RS2 Brane Cosmology Model. http://arxiv.org/abs/0812.4197.
- A. BACHELOT. The Klein-Gordon Equation in Anti-de Sitter Cosmology. http://arxiv.org/abs/1010.1925
- A. BACHELOT. New Dynamics in the Anti-De Sitter Universe AdS^5. http://arxiv.org/pdf/1112.6138v2
- A. BACHELOT. On the Klein-Gordon equation near a De Sitter brane in an Anti-de Sitter bulk, arXiv:1402.1071v3
- A. BACHELOT. Waves in the Witten Bubble of Nothing and the Hawking Wormhole, arXiv:1601.03682
- A. BACHELOT-MOTET, A.BACHELOT. Waves on accelerating dodecahedral universes, arXiv:1609.00806
- A. BACHELOT. Wave asymptotics at a cosmological time-singularity, arXiv:1806.01543
We prove the Scattering Operator for the Non Linear Klein-Gordon Equation with interacting $q(x)u^3$ determines the coupling potential $q(x)$.
Given a square matrix $(f_{j,k}(x))$ compatible with the system $A(x, D)=\sum_{i=1}^n A_i(x)D_i$ we define $\sum_{1\leq j, k\leq N} f_{j,k}(x)u_iu_k$ on Sobolev or Besov subspaces. New results on compensated compacteness are obtained for systems with variable coefficients.
We consider the Cauchy problem for certain semilinear hyperbolic systems such as the Schroedinger-Klein-Gordon equations and the coupled Schroedinger equations of the type $-i\psi\sb t-\Delta \psi =F(\psi,\phi);\ -i\phi\sb t-\Delta \phi =G(\psi,\phi)$ in three space dimensions. - In contrast to former results the nonlinearities are general enough to exclude the use of energy conservation but allow the use of charge conservation. Essentially they have to grow quadratically. Under these assumptions a global existence and uniqueness result is proven e.g. in the space $C\sp 0({\bbfR},L\sp 4({\bbfR}\sp 3)\cap L\sp 2({\bbfR}\sp 3)).$ The necessary a-priori-bounds can be given by use of the well-known $L\sp p-L\sp{p'}$-estimates for the solutions of the linear problem. - In the second part of the paper the Dirac-Klein-Gordon system with a generalization of the Yukawa coupling is considered and a local existence and uniqueness result is proven.
We study the continuity of the solution of the Klein-Gordon equation with respect to the mass. We prove the convergence in $L\sp q(R\sb t\times R\sp n\sb x)$ of the solution of the inhomogeneous Klein-Gordon equation to the solution of the wave equation, with same initial data, when the mass tends to 0; we use this result to solve the inverse scattering problem for the equation $$ \square u+m\sp 2u=\sum\sb{k\ge 1}q\sb k(x)\vert u\vert\sp{2k}u.$$
Given a hyperbolic system I $\partial\sb t\psi -\sum\sp{n}\sb{i=1}A\sb i\partial\sb x\psi$ and a sesquilinear form f, $I(t)=\int f(\psi (t,x),\psi (t,x))dx$ tends to zero when $\vert t\vert \to \infty$ for any finite energy solution $\psi$ if and only if f is compatible with the system in the sense of B. Hanouzet and J. L. Joly, i.e. f(Ker$\sum\sp{n}\sb{i=1}\xi\sb iA\sb i)=0$ for a.e. $\xi \in {\bbfR}\sp n$. If n is odd and the multiplicity of the system is constant, $I(t)=0$ after a finite time for solutions having initial data with compact support. We also study the hermitian systems I $\partial\sb t\psi -\sum\sp{n}\sb{i=1}A\sb i\partial\sb{x\sb i}\psi +iB\psi$. We prove the equipartition of energy for the hyperbolic equations $\partial\sp 2\sb{tt}\psi -A\sp 2(d)\psi +B\sp 2\psi =0$, the wave equation, the elastic waves in anisotropic media, the magneto-elastic waves, the Klein-Gordon equation, Maxwell's equations, the Dirac system and the Neutrino equation.
We prove the existence of the scattering operator for the wave equation with a potential which is periodic in time and has compact support in space, in dimension greater than or equal to 3, provided the energy is uniformly bounded. The key result is the decay of the local energy. We get strong convergence by using the compactness of the local evolution operator, derived from a microlocal analysis of the propagation of singularities. In the case where the dimension is odd, the decay is exponential for initial data: i) with compact support and ii) included in a subspace of finite codimension. We give some sufficient conditions for the boundedness of the energy by studying the spectrum of the local evolution operator. We extend these results to first order hermitian systems with arbitrary multiplicity and with a periodic potential such as the Dirac system in a periodic electromagnetic field.
We consider a massive Dirac system quadratically coupled with a wave equation in three space dimensions. The global Cauchy problem is well posed if the nonlinearities satisfy some algebraic conditions related to the Lorentz-invariance, the null condition and the compatibility of a sesquilinear form with the Dirac system. The fundamental example is the pseudoscalar Yukawa model of nuclear forces. We use some $L\sp 2-L\sp{\infty}$ estimates in Sobolev spaces associated with the Lorentz metric, for the Dirac equation with a potential. We establish the completeness of the scattering operator for an electron in a free electromagnetic field.
We prove the existence of some Global Solutions, with Large Energy, of relativistic Dirac-Klein-Gordon systems with quadratic coupling and cubic autointeractions in Minkowski space.
We prove the existence of global solutions of the Cauchy problem for the nonlinear massless Dirac equation, without assumptions on the ``size'' of the smooth initial data and only assuming the smallness of the Chiral invariant. Moreover we study the asymptotic behaviour of the solution, namely the equipartition of energy and the decay of Lorentz-invariant products.
The paper is devoted to the electromagnetic scattering by a spherical black-hole in the Schwarzschild spacetime. Some wave operators are introduced, yielding an electromagnetic field far from the black-hole ($W\sb 0\sp \pm$) and near the Schwarschild radius ($W\sp \pm\sb 1$). The existence of the scattering operator is proved by the Birman-Kato method. The asymptotic completeness of $W\sp +\sb 1$ implies that near the horizon, the fields of finite redshifted energy are described by ingoing plane waves. In the Kruskal universe, the same argument for $W\sp \pm\sb 0$ and $W\sp +\sb 1$ allows the definition of the solution on the future horizons. The scattering operator can be approximated by putting the impedance condition on the stretched horizon, a fact that justifies the Membrane Paradigma {\it D. A. MacDonald} and {\it W. M. Suen}, Phys. Rev. D 32, 848-871 (1985)].
We prove the existence and asymptotic completeness of the Wave Operators describing the asymptotic behaviours of the electromagnetic field at the Black-Hole Horizon and at the Cosmological Horizon.
We solve the global Cauchy problem for the non linear Klein-Gordon equation outside a spherical Black Hole. On the Black Hole Horizon the field satisfies the impedence condition of T. Damour. If the space time is asymptotically flat, the massless fields satisfy the Sommerfeld condition at infinity.
This paper is devoted to the theoretical and computational investigations of the scattering frequencies of scalar, electromagnetic, gravitational waves around a spherical black hole. We adopt a time dependent approach: construction of wave operators for the hyperbolic Regge-Wheeler equation; asymptotic completeness; outgoing and incoming spectral representations; meromorphic continuation of the Heisenberg matrix; approximation by dumping and cut-off of the potentials and interpretation of the semigroup $\bbfZ(t)$ in the framework of the membrane paradigm. We develop a new procedure for the computation of the resonances by the spectral analysis of the transient scattered wave, based on Prony's algorithm.
We prove the strong asymptotic completeness of the wave operators, classic at the horizon and Dollard-modified at infinity, describing the scattering of a massive Klein-Gordon field by a Schwarzschild black hole. The scattering operator is unitarily implementable in the Fock space of free fields.
We solve the problem of diffraction of an electromagnetic wave by an absorbing body using a boundary integral method in time-domain directly. We prove the existence and uniqueness of the solution of this problem. We obtain the continuity and a relation of coercivity for the associated time-dependent formulation in this time functional framework. The discret approximation of the variational formulation leads to a stable marching-in-time scheme.
Abstract. We consider the problem of scattering transient waves, by an inhomogeneous object, in Èsup2(dfo()3). The idea consists in coupling a finite element resolution in the volume including all inhomogeneities, with an integral equation expressing the perfectly transparent condition on the boundary. This condition, which comes from Kirchhoff's formula, introduces single and double layer retarded potentials. The integral operator can be studied, according to the Ha-Duong approach, with Fourier-Laplace transform. That leads us to the associated harmonic problem, for which we prove existence and uniqueness. We also construct another formulation of the problem which satisfies properties of coercivity. The discretization of the first time-space variational formulation conducts to a time-stepping scheme, for which we present numerical computations.
We study the scattering of a scalar field, massive or massless, by a collapsing spherical star. The main point of interest is the infinite Doppler effect measured by an observer, at rest in Schwarzschild coordinates. We construct the functional framework associated with this phenomenon, and we prove the existence and strong asymptotic completeness of the wave operators describing the scattering of the field by the space-time curvature, and the asymptotically characteristic moving boundary of the star.
We prove the decay of the local energy of an acoustic wave, scattered by an inhomogeneous obstacle. We establish that this wave is eventually outgoing.
We prove in the case of the Klein-Gordon quantum field, the emergence of the Hawking-Unruh state at the future Black-Hole horizon created by a spherical gravitational collapse. This is a rigorous proof of the famous result by S. Hawking on the Quantum Radiation near the Black-Hole Horizon.
We present and analyse a parallel algorithm for solving time convolution equations. We consider applications to the marching-in-time-scheme of the Time Dependent Integral Method in electromagnetic scattering.
We consider a spherical star, stationary in the past, collapsing to a Black-Hole in the future. Assuming the quantum state of a Klein-Gordon field to be the Fock vacuum in the past, we prove that an observer at rest in the Schwarzschild coordinates, will measure a thermal state with the Hawking temperature, at the last time of the gravitational collapse.
We investigate the quantum state of the Dirac field at the horizon of a charged black-hole formed by a spherical gravitational collapse. We prove this state satisfies a KMS condition with the Hawking temperature and the chemical potential associated with the mass and the charge of the black-hole. Moreover, the fermions with charge of same sign to that of the black-hole are emitted more readily than those of opposite charge. It is a spontaneous loss of charge of the black-hole due to the quantum vacuum polarization.
We investigate the coupling of finite elements and retarded potentials for an electromagnetic scattering problem by an inhomogeneous obstacle. We construct several variational formulations and we investigate the stability and the convergence of these methods.
We introduce a class of four dimensional Lorentzian manifolds with closed curves of null type or timelike. We investigate some global problems for the wave equation: uniqueness of solution with data on a changing type hypersurface; existence of resonant states; scattering by a violation of the chronology; global Cauchy problem and asymptotic completeness of the wave operators for the chronological but non-causal metrics.
We develop the scattering theory for the charged Klein-Gordon equation on $\RR_t\times\RR_x$, when the electrostatic potential $A(x)$ has different asymptotics $a^{\pm}$ as $x\rightarrow\pm\infty$. In this case, the conserved energy is not positive definite (Klein Paradox). We construct the spectral representation for the harmonic equation. Since $a^+\neq a^-$, the distorded Fourier transform has to be defined on weighted $L^2$-spaces, and it can appear spectral quantities of new type, that are neither eigenvalue, nor resonance. These so called "hyperradiant modes" are real singularities of the Green function, and lead to polynomialy increasing solutions in time. We investigate the asymptotic behaviours of the solutions as $t\rightarrow\pm\infty$, and we establish the existence of a Scattering Operator of which the symbol has a norm strictly larger than 1, for the frequencies in $(a^-,a^+)$. We apply these results to the DeSitter-Reissner-Nordstr{\o}m metric, to rigorously justify the notion of superradiance of the charged black-holes.
We investigate the global Cauchy problem for a class of semilinear hyperbolic systems where the interaction can be non local in space and time. We establish global existence theorems for the initial value problem when the non linearity is dissipative in a weak sense, and satisfies the causality condition. The argument is abstract and the technique is based on the non-linear resolvent. We apply these results to get low regularity global solutions of several models for relativistic field theory : the Dirac-Maxwell-Klein-Gordon system, and the Thirring model on the Minkowski space-time $\RR^{1+1}$; the Dirac-Klein-Gordon system on Schwarzschild type manifolds, or outside a star undergoing a gravitational collapse to a black-hole.
In this paper we present a survey of some recent results on the global existence and the asymptotic behaviour of waves, when the conserved energy is not definite positive. This unusual situation arises in important cosmological models of the General Relativity where the gravitational curvature is very strong. We consider the case of the closed time-like curves (violation of the causality) [1], and the charged black-holes (superradiance) [3].
We investigate the global solutions of the Dirac equation on the Anti-de-Sitter Universe. Since this space is not globally hyperbolic, the Cauchy problem is not, a priori, well-posed. Nevertheless we can prove that there exists unitary dynamics, but its uniqueness crucially depends on the ratio beween the mass $M$ of the field and the cosmological constant $\Lambda > 0$ : it appears a critical value, $\Lambda /12$, which plays a role similar to the Breitenlohner-Freedman bound for the scalar fields. When $M^2 \geq \Lambda/12$ there exists a unique unitary dynamics. In opposite, for the light fermions satisfying $M^2 < \Lambda/12$, we construct several asymptotic conditions at infinity, such that the problem becomes well-posed. In all the cases, the spectrum of the hamiltonian is discrete. We also prove a result of equipartition of the energy.
We study the wave equation for the gravitational fluctuations in the Randall-Sundrum brane cosmology model. We solve the global Cauchy problem and we establish that the solutions are the sum of a slowly decaying massless wave localized near the brane, and a superposition of massive dispersive waves. We compute the kernel of the truncated resolvent. We prove some $L^1-L^{\infty}$, $L^2-L^{\infty}$ decay estimates and global $L^p$ Strichartz type inequalities. We develop the complete scattering theory : existence and asymptotic completness of the wave operators, computation of the scattering matrix, determination of the resonances on the logarithmic Riemann surface.
This paper deals with the Klein-Gordon equation on the Poincar\'e chart of the 5-dimensional Anti-de Sitter universe. When the mass $\mu$ is larger than $-\frac{1}{4}$, the Cauchy problem is well posed despite the loss of global hyperbolicity due to the time-like horizon. We express the finite energy solutions in the form of a continuous Kaluza-Klein tower and we deduce a uniform decay as $\mid t\mid^{-\frac{3}{2}}$. We investigate the case $\mu=\frac{\nu^2-1}{2}$, $\nu\in\NN^*$, which encompasses the gravitational fluctuations, $\nu=4$, and the electromagnetic waves, $\nu=2$. The propagation of the wave front set shows that the horizon acts like a perfect mirror. We establish that the smooth solutions decay as $\mid t\mid^{-2-\sqrt{\mu+\frac{1}{4}}}$, and we get global $L^p$ estimates of Strichartz type. When $\nu$ is even, there appears a lacuna and the equipartition of the energy occurs at finite time for the compactly supported initial data, although the Huygens principle fails. We address the cosmological model of the negative tension Minkowski brane, on which a Robin boundary condition is imposed. We prove the hyperbolic mixed problem is well-posed and the normalizable solutions can be expanded into a discrete Kaluza-Klein tower. We establish some $L^2-L^{\infty}$ estimates in suitable weighted Sobolev spaces.
This paper deals with the propagation of the gravitational waves in the Poincar ́e patch of the 5-dimensional Anti-de Sitter universe. We construct a large family of unitary dynamics with respect to some high order energies that are conserved and positive. These dynamics are associated with asymptotic conditions on the conformal time-like boundary of the universe. This result does not contradict the statement of Breitenlohner-Freedman that the hamiltonian is essentially self- adjoint in L2 and thus accordingly the dynamics is uniquely determined. The key point is the introduction of a new Hilbert functional framework that contains the massless graviton which is not normalizable in L2. Then the hamiltonian is not essentially self-adjoint in this new space and possesses a lot of different positive self-adjoint extensions. These dynamics satisfy a holographic principle : there exists a renormalized boundary value which completely characterizes the whole field in the bulk.
In this paper we investigate the Klein-Gordon equation in the past causal domain of a De Sitter brane imbedded in a Anti-de Sitter bulk. We solve the global mixed hyperbolic problem. We prove that any finite energy solution can be expressed as a Kaluza-Klein tower that is a superposition of free fields in the Steady State Universe, of which we study the asymptotic behaviours. We show that the leading term of a gravitational fluctuation is a massless graviton, i.e. the De Sitter brane is linearly stable.
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Alain
BACHELOT
Professor Emeritus
University of Bordeaux
Address :
Institute of Mathematics of Bordeaux
351, cours de la Libération
F-33405 TALENCE cedex
Tél : (33 5) / (05) 40 00 60 63
Fax. (33 5) / (05) 40 00 26 26
E-mail : alain.bachelot@u-bordeaux.fr
Version
française
Degrees in mathematics :
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