Alain BACHELOT
Université de Bordeaux

• Fonctions actuelles : Professeur des universités (CEx2).
• Thèmes de recherche

• Equations aux dérivées partielles hyperboliques
• Théorie de la diffusion
• Champs quantiques en relativité générale
• Trous noirs
• Cosmologie branaire
• Electromagnétisme numérique
• Enseignement : visiter cette page.

Institut de Mathématiques de Bordeaux
351, cours de la Libération
F-33405 TALENCE cedex

bâtiment A33, bureau 283

Tél : (33 5) / (05) 40 00 60 63
Fax. (33 5) / (05) 40 00 26 26
E-mail :
alain.bachelot@u-bordeaux.fr

Pour plus de détails :
Pour visiter l'Université de Bordeaux, l'UF mathématiques et interactions, la Bibliothèque, l'Institut de Mathématiques, l'Equipe EDP et physique mathématique, ou revoir Newton.

Dernière mise à jour, 6 juin 2018.

• Diplômes en mathématiques :
Agrégation (1979)
Doctorat (1981)
Habilitation (1988)
• Distinctions :
Prix "Institut Henri Poincaré / Gauthier-Villars" en physique théorique, 1998.
Featured Review in Mathematical Reviews, 1999.
Promotion à la classe exceptionnelle des professeurs des universités par le CNU, 2009.
Chevalier dans l'Ordre des Palmes Académiques, 2013.
Promotion au dernier échelon de la classe exceptionnelle par le CNU, 2014.

• Positions :

à l'Ecole Normale Supérieure de Rabat-Souissi, Maroc :

Assistant agrégé (1982-1984)

à l'Université de Bordeaux :

Assistant agrégé (1984)
Maître de conférences (1988)
Professeur des universités, seconde classe (1990)
Professeur des universités, première classe (1996)
Professeur des universités, classe exceptionnelle, 1° échelon (2009-2014)
Professeur des universités, classe exceptionnelle, 2ème échelon (2014-.)

Conseiller scientifique au CEA (2000-2008).

• Enseignement : visiter cette page.
• Président de la Commission de spécialistes 26° section (1991-1994)
• Directeur de l'Ecole doctorale de mathématiques et informatique de Bordeaux (1995-1998)
• Membre élu du Conseil National des Universités, membre du bureau (1996-1998)
• Membre nommé au Conseil National des Universités (1999-2003)
• Directeur de l'UMR CNRS 5466 de mathématiques appliquées (1999-2002) et du LRC CEA M03 (2000-2002)
• Membre (2002-2007) et Secrétaire (2003-2007) du Conseil Scientifique de la SMF
• Membre du Comité pour l'égalité professionnelle hommes/femmes - Ministère de la Recherche (2006-2010)
• Président de la Commission Consultative 26° section (2008-2010)
• Responsable de la licence d'ingénierie mathématique (2007-2010)
• Responsable du Master MIMSE (2010-2015)
• Responsable de l'équipe "EDP et physique mathématique" (2010-2014)
• Thèmes de recherche
• Equations hyperboliques linéaires et non-linéaires des champs dans l'espace de Minkowski
• problème de diffusion inverse pour l'équation non linéaire de Klein-Gordon (1981)
• problème de Cauchy global à données peu régulières pour des systèmes de Klein-Gordon-Schrödinger (1983)
• produit de distributions vectorielles dans les espaces de Besov (avec B. Hanouzet, 1983)
• partition de l'énergie des solutions d'un système hyperbolique (1985)
• diffraction par un potentiel $t$-périodique pour des systèmes hyperboliques (avec V. Petkov, 1987)
• problème de Cauchy global à petites données pour les systèmes de Dirac-Klein-Gordon Lorentz-invariants (1988)
• existence de solutions globales de grande énergie pour ces systèmes (1988-1989)
• problème de Cauchy global pour des systèmes hyperboliques à nonlinaérité non locale (2005)
• Théorie des champs en Relativité Générale, Trous Noirs, Cosmologie branaire (Thème principal)
• système de Maxwell : existence et complétude asymptotique des opérateurs d'onde en métrique de Schwarzschild (1991) et de De Sitter-Schwarzschild (1992)
• équation de Regge-Wheeler : existence de la matrice de diffraction S, prolongement méromorphe de S, calcul effectif des résonances d'un trou noir sur CRAY 2 du CCVR (avec A. Bachelot-Motet., 1993)
• équation non linéaire de Klein-Gordon en métrique de Schwarzschild : problème de Cauchy global, asymptotiques (avec J-P. Nicolas, 1993)
• équation linéaire de Klein-Gordon en métrique de Schwarzschild : complétude asymptotique des opérateurs d'onde Dollard-modifiés, seconde quantification du champ, opérateur de scattering quantique (1994)
• diffusion d'un champ de Klein-Gordon par une étoile en effondrement gravitationnel (1995)
• polarisation du vide quantique à l'horizon d'un trou noir (radiation Hawking) (1997)
• effet Hawking (1998)
• émission de particules et d'antiparticules par un trou noir chargé (1999)
• diffusion des ondes par une violation de la causalité (2002)
• superradiance des trous noirs chargés (2004)
• champs de Dirac-Klein-Gordon globaux dans des espaces-temps trous noirs (2005)
• champs de Dirac globaux dans l'univers Anti-de Sitter (2007)
• propagation et diffusion en cosmologie branaire RS2 (2008)
• équation de Klein-Gordon en cosmologie Anti-de Sitter AdS5 (2010)
• nouvelles conditions aux bord de l'univers Anti-de Sitter (2012)
• stabilité linéaire d'une brane de De Sitter (2014)
• bulle de néant de Witten, trou de vers de Hawking (2016)
• univers dodécaédriques en accélération (avec A. Bachelot-Motet, 2016)
• asymptotique d'ondes à une singularité cosmologique temporelle (2018)
• Méthodes numériques en électromagnétisme
• potentiels retardés pour un obstacle conducteur en 3D+1 (avec A. Pujols, 1991)
• couplage éléments finis / potentiels retardés en 3D+1 (avec V. Lubet, 1994)
• potentiels retardés pour un obstacle absorbant en 3D+1 (avec V. Lange 1995)
• parallélisation d'un code par potentiels retardés 3D+1 (avec P. Charrier, 1997)
• diffraction par des obstacles hétérogènes en 3D+1 (avec L. Bounhoure, 1998)
• couplage méthode multipoles / discrétisation microlocale / ordre élevé (avec E. Darrigrand, L. Gatard, K. Mer, 2002-2007)
• méthode multipoles pour l'équation des radiosités (avec J. Morice, K. Mer, 2007)

PUBLICATIONS

Comptes-rendus de l'Académie des Sciences de Paris / Academy of Sciences of Paris

A. BACHELOT. Problème inverse de diffusion non linéaire. C. R. Acad. Sc. Paris, Série I, 293 : 121-124, 1981.

A. BACHELOT. Problème de Cauchy pour des systèmes de Klein-Gordon-Schrödinger. C. R. Acad. Sc. Paris, Série I, 296 : 525-528, 1983.

A. BACHELOT, B. HANOUZET. Applications bilinéaires compatibles avec un système différentiel à coefficients variables. C. R. Acad. Sc. Paris, Série I, 299 : 543-546, 1984.

A. BACHELOT. Equipartition de l'énergie pour les systèmes hyperboliques et formes compatibles. C. R. Acad. Sc. Paris, Série I, 301 : 573-576, 1985.

A. BACHELOT, V. PETKOV. Existence de l'opérateur de diffusion pour l'équation des ondes avec un potentiel périodique en temps. C. R. Acad. Sc. Paris, Série I, 303 : 671-673, 1986.

A. BACHELOT. Opérateur de diffraction pour le système de Maxwell en métrique de Schwarzschild. C. R. Acad. Sc. Paris, Série I, 312 : 93-96, 1991.

A. BACHELOT, A. PUJOLS. Equations intégrales espace-temps pour le système de Maxwell. C. R. Acad. Sc. Paris, Série I, 314 : 639-644, 1992.

A. BACHELOT, A. MOTET-BACHELOT. Les pôles de résonance de la métrique de Schwarzschild. C. R. Acad. Sc. Paris, Série I, 316 : 795-798, 1993.

A. BACHELOT, J-P. NICOLAS. Equation non linéaire de Klein-Gordon dans des métriques de type Schwarzschild. C. R. Acad. Sc. Paris, Série I, 316 : 1047-1050, 1993.

A. BACHELOT. Opérateur de diffusion classique et quantique pour l'équation de Klein-Gordon en métrique de Schwarzschild. C. R. Acad. Sc. Paris, Série I, 319 : 41-44, 1994.

A. BACHELOT. Diffusion d'un champ scalaire par un effondrement gravitationnel. C. R. Acad. Sc. Paris, Série I, 321 : 1329-1332, 1995.

A. BACHELOT. La radiation Hawking à l'horizon d'un trou noir. C. R. Acad. Sc. Paris, Série I, 324 : 855-860, 1997.

A. BACHELOT. L'effet Hawking. C. R. Acad. Sc. Paris, Série I, 325 : 1229-1234, 1997.

A. BACHELOT. Création de fermions à l'horizon d'un trou noir chargé. C. R. Acad. Sc. Paris, Série I, 330 : 28-34, 2000.

A. BACHELOT. Diffusion des ondes par une violation de la causalité. C. R. Acad. Sc. Paris, Série I, 333 : 1065-1068, 2001.

A. BACHELOT, E. DARRIGRAND, K. MER-NKONGA. Coupling of a Multilevel fast Multipole and a Microlocal Discretization for the 3-D Integral Equations of Electromagnetism. C. R. Acad. Sc. Paris, Série I, 336 : 505-510, 2003.

A. BACHELOT. Paradoxe de Klein pour l'équation de Klein-Gordon chargée : superradiance et opérateur de diffusion. C. R. Acad. Sc. Paris, Série I, 339 : 345-350, 2004.

A. BACHELOT. The Dirac Equation on the Anti-De-Sitter Universe. C. R. Acad. Sc. Paris, Série I, 345 : 435-440, 2007.

A. BACHELOT. Scattering by a Minkowski Brane World. C. R. Acad. Sc. Paris, Série I,  347 : 1243-1248, 2009.

A. BACHELOT. On the wave propagation in the anti-de Sitter cosmology. C. R. Math. Acad. Sci. Paris, Série I,  349 : 47-51, 2011.

A. BACHELOT. New boundary conditions on the time-like conformal infinity of the anti-de Sitter universe. C. R. Math. Acad. Sci. Paris, Série I, 350 : 359-364, 2012.

A. BACHELOT. Wave fluctuations near a De Sitter brane in an ti-de Sitter universe. C. R. Math. Acad. Sci. Paris, Série I, 354 : 19-25, 2016.

Articles dans des revues à comité de lecture / Articles in Journal with referee

A. BACHELOT. Problème de Cauchy pour des systèmes hyperboliques semi-linéaires. Ann. Inst. Henri Poincaré, Analyse non linéaire, 1(6) : 453-478, 1984.

A. BACHELOT. Convergence dans $L^p(\BbbR^n+1)$ de la solution de l'équation de Klein-Gordon vers celle de l'équation des ondes. Ann. Fac. Sci. Toulouse, 8(1) : 37-60, 1986.

A. BACHELOT. Equipartition de l'énergie pour les systèmes hyperboliques et formes compatibles. Ann. Inst. Henri Poincaré - Physique théorique, 46(1) : 45-76, 1987.

A. BACHELOT, V. PETKOV. Existence des opérateurs d'ondes pour les systèmes hyperboliques avec un potentiel périodique en temps. Ann. Inst. Henri Poincaré - Physique théorique, 47(4) : 383-428, 1987.

A. BACHELOT. Problème de Cauchy global pour des systèmes de Dirac-Klein-Gordon. Ann. Inst. Henri Poincaré - Physique théorique, 48(4) : 387-422, 1988.

A. BACHELOT. Global existence of large amplitude solutions for non linear massless Dirac equations. Portugaliae Math., 46(5) : 455-473, 1989.

A. BACHELOT. Gravitationnal Scattering of Electromagnetic Field by Schwarzschild Black Hole. Ann. Inst. Henri Poincaré - Physique théorique, 54(3) : 261-320, 1991.

A. BACHELOT, A. MOTET-BACHELOT. Les résonances d'un trou noir de Schwarzschild. Ann. Inst. Henri Poincaré - Physique théorique, 59(1) : 3-68, 1993.

A. BACHELOT.  Asymptotic Completeness for the Klein-Gordon Equation on the Schwarzschild Metric. Ann. Inst. Henri Poincaré - Physique théorique, 61(4) : 411-441, 1994.

A. BACHELOT.  Scattering of Scalar Fields by Spherical Gravitational Collapse. J. Math. Pures Appl., 76(2) : 155-210, 1997.

A. BACHELOT, A. PUJOLS. Décroissance de l'énergie locale d'une onde diffusée par un obstacle inhomogène. Rev. Roumaine Math. Pures et Appl., 41(7-8) : 451-459, 1997.

A. BACHELOT. Quantum Vacuum Polarization at the Black-Hole Horizon. Ann. Inst. Henri Poincaré - Physique théorique, 67(2) : 181-222, 1997.

A. BACHELOT. The Hawking Effect. Ann. Inst. Henri Poincaré - Physique théorique, 70(1) : 41-99, 1999.

A. BACHELOT. Creation of Fermions at the Charged Black-Hole Horizon. Ann. Henri Poincaré, 1 : 1043-1095, 2000.

A. BACHELOT, L. BOUNHOURE, A. PUJOLS. Couplage éléments finis-potentiels retardés pour la diffraction électromagnétique par un obstacle inhomogène. Numer. Math., 89 : 257-306, 2001.

A. BACHELOT. Global properties of the wave equation on non globally hyperbolic manifolds. J. Math. Pures Appl., 81 : 35-65, 2002.

A. BACHELOT. Superradiance and Scattering of the Charged Klein-Gordon Field by a Steplike Electrostatic Potential. J. Math. Pures Appl., 83 : 1179-1239, 2004.

A. BACHELOT. Global Cauchy problem for semilinear hyperbolic systems with non-local nonlinearities. Applications to Dirac equations. J. Math. Pures Appl., 86 : 201-236, 2006.

A. BACHELOT. Global waves with non positive energy in general relativity. Serdica Math. J., 34 : 127-154, 2008.

A. BACHELOT. The Dirac System on the Anti-de Sitter Universe . Comm. Math. Phys. 283 : 127-167, 2008.

A. BACHELOT. Wave Propagation and Scattering for the RS2 Brane Cosmology Model. J. Hyperbolic Differ. Equ., 6(4) : 809-861, 2009.

A. BACHELOT.  The Klein-Gordon Equation in Anti-de Sitter Cosmology, J. Math. Pures Appl., 96 : 527-554, 2011.

A. BACHELOT. New Dynamics in the Anti-de Sitter Universe AdS^5. Comm. Math. Phys., 320, 723-759, 2013.

A. BACHELOT. J. Math. Pures Appl., 105, 165-197, 2016.

A. BACHELOT. Waves in the Witten Bubble of Nothing and the Hawking Wormhole. Comm. Math. Phys., 351(2), 599–651, 2017. To read online

A. BACHELOT-MOTET, A.BACHELOT. Class. Quantum Gravity, 35(5), 055010 (39pp), 2017.

Conférences publiées avec comité de lecture / Proceedings with referee

A. BACHELOT. Inverse scattering problem for the nonlinear Klein-Gordon equation. In J-I. Diaz J. Herdandez C. Bardos, A. Damlamian, editor, Contributions to nonlinear partial differentials equations, volume 89 of Research Notes in Mathematics, pages 7-15. Pitman, 1983.

A. BACHELOT. Problèmes de Cauchy pour des systèmes hyperboliques semi-linéaires. In Equations aux dérivées partielles, Saint Jean de Monts, pages VIII.1-VIII.5, 1983.

A. BACHELOT, V. PETKOV. Existence de l'opérateur de diffusion pour l'équation des ondes avec un potentiel périodique en temps. In J-L. Lions, editor, Nonlinear partial differentials equations and their applications - Collège de France seminar, volume 181 of Research Notes in Mathematics, pages 13-27. Pitman, 1983.

A. BACHELOT. Equipartition de l'énergie pour les systèmes hyperboliques et formes compatibles. In Equations aux dérivées partielles, Saint Jean de Monts, pages XIII.1-XIII.8, 1986.

A. BACHELOT. Opérateurs de convolution définis à partir d'une forme quadratique. In Equations aux dérivées partielles, Saint Jean de Monts, pages XVI.1-XVI.8, 1982.

A. BACHELOT. Solutions globales des systèmes de Dirac-Klein-Gordon. In Equations aux dérivées partielles, Saint Jean de Monts, pages XV.1-XV.10, 1987.

A. BACHELOT. Global existence of large amplitude solutions for Dirac-Klein-Gordon systems in Minkowski space. In Non linear Hyperbolic Problems, volume 1402 of Lecture Notes in Math., pages 99-113. Springer Verlag, 1989.

A. BACHELOT. Global solutions for nonlinear Dirac equations. In Integrables Systemes and applications, volume 342 of Lecture Notes in Physics, pages 1-11. Springer Verlag, 1989.

A. BACHELOT. Opérateur de diffraction pour le système de Maxwell en métrique de Schwarzschild. In Equations aux dérivées partielles, Saint Jean de Monts, pages III.1-III.11, 1990.

A. BACHELOT. Scattering by black-hole for electromagnetic field. In Inverse Methods in Action, pages 174-181. Springer Verlag, 1990.

A. BACHELOT. Scattering operator for Maxwell equations outside Schwarzschild black-hole. In Integral equations and Inverse problems, volume 235 of Research Notes in Mathematics, pages 38-48. Pitman, 1991

A. BACHELOT. Scattering for Maxwell equations on Schwarzschild metric. In "International Congress of Mathematicians", page 209. Kyoto, 1990.

A. BACHELOT. Scattering of electromagnetic field by de Sitter-Schwarzschild black-hole. In Non linear hyperbolic equations and field theory, volume 253 of Research Notes in Mathematics, pages 23-35. Pitman, 1992.

A. BACHELOT, A. PUJOLS. Boundary integral equation method in time domain for Maxwell's system. In 10th International Conference on Computing Methods in Applied Sciences and Engineering, pages 197-206. Nova Science Publishers, 1992.

A. BACHELOT. Calcul des résonances d'un trou noir. In Ecole des ondes, pages 1-28. INRIA, 1993.

A. BACHELOT. La diffraction en métrique de Schwarzschild : complétude asymptotique et résonances. In Séminaire X-EDP, pages VIII.1-VIII.13. Ecole Polytechnique, 1993.

A. BACHELOT,  A. MOTET-BACHELOT. Scattering resonances for Schwarzschild black-hole. In Oliveri Donato, editor, Nonlinear Hyperbolic Problems : Theoretical, Applied, and Computational Aspects of Wave Propagation Phenomena, volume 43 of Vieweg Notes, pages 33-40, 1993.

A. BACHELOT, V. LANGE. Time dependent integral method for Maxwell's system. In G. Cohen, editor, Third International Conference on Mathematical and Numerical Aspects of Wave Propagation Phenomena, pages151-159. SIAM-INRIA, 1995.

A. BACHELOT, V. LANGE. Time dependent integral method for Maxwell's system with impedance boundary condition. In 10th International Conference on Boundary Element Technology, BETECH95, pages 137-144. Computational Mechanics Publications, 1995.

A. BACHELOT, V. LUBET. On the coupling of boundary element and finite element methods for a time problem. In G. Cohen, editor, Third International Conference on Mathematical and Numerical Aspects of Wave Propagation Phenomena, pages 130-139. SIAM-INRIA, 1995.

A. BACHELOT. Diffusion classique et quantique par un trou noir en formation. In Séminaire X-EDP, pages XV.1-XV.18. Ecole Polytechnique, 1996.

A. BACHELOT, P. CHARRIER, A. PUJOLS, D. ROUART. Parallel Algorithm for Solving Time Convolution Equations and Application to CEM. In 8th SIAM Conference on Parallel Processing for Scientific Computing, Minneapolis 1997, (CD-ROM). SIAM, 1997.

A. BACHELOT. Scattering operator for the Klein-Gordon equation on the Schwarzschild metric. In International Conference on Non Linear Evolution Partial Differential Equations, Beijing 93, AMS/IP Studies in Advanced Math., volume 3, 1997.

A. BACHELOT. L'effet Hawking. in Séminaire X-EDP, Ecole Polytechnique 1997.

A. BACHELOT, J-P. RAOULT. Mathématiques et entreprises. In Mathématiques A Venir, SMF-SMAI, Gazette des mathématiciens, supp. au n°75, 1997.

A. BACHELOT, L. BOUNHOURE, A. PUJOLS. Coupling of finite elements and retarded potentials for an electromagnetic scattering problem by an inhomogeneous obstacle. In Fourth International Conference on Mathematical and Numerical Aspects of Wave Propagation Phenomena, Golden, Colorado, SIAM, 1998.

A. BACHELOT. Wave equation and causality violation. in Séminaire X-EDP, Ecole Polytechnique 2001.

A. BACHELOT, E. DARRIGRAND, K. MER-NKONGA. Coupling of a Fast Multipole Method and a Microlocal Discretization for Integral Equations of Electromagnetism. In  "WAVES 2003, Jyväskylä, Finlande, SIAM-INRIA, Springer Verlag, 2003.

A. BACHELOT. Paradoxe de Klein et diffusion pour l'équation de Klein-Gordon chargée. in Séminaire X-EDP, Ecole Polytechnique 2004.

L. GATARD, A. BACHELOT, K. MER-NKONGA. High order boundary integral methods for Maxwell²s equations: coupling of microlocal discretization and fast multipole methods. In "ENUMATH 05", Springer-Verlag, 2005.

J. MORICE, K. MER-NKONGA, A. BACHELOT.  Fast multipole method for solving the radiosity equation. In "ENUMATH 05", Springer-Verlag, 2005.

J. MORICE, K. MER-NKONGA, A. BACHELOT. Comparison of different fast multipole methods for solving the radiosity equation. In "Advances in Boundary Element Techniques VII", M.H. Aliabadi, B. Gatmiri, A. Sellier edts., 2006.

J. MORICE, K. MER-NKONGA, A. BACHELOT. Numerical comparison of different fast multipole method for the radiosity equation. In "WAVES 07", Springer, 2007.

Publications d'intérêt général / Miscellanea

A. BACHELOT, C. VIDAL Edts. La recherche en sciences : Morceaux choisis. Publications de l'Université Bordeaux-1, 1994.

A. BACHELOT. J-P. RAOULT. Mathématiques et entreprises. In "Mathématiques A Venir", SMF-SMAI, Gazette des mathématiciens, supp. au numéro 75, 1997.

A. BACHELOT,  R. DIENG-KUNTZ, E. DUBOIS-VIOLETTE, L. HEMIDY, C. HERMANN, M. IMBERT, J. MASSOT, L. NURIT, F. THEBAUD. Rapport sur l'égalité professionnelle entre les femmes
et les hommes dans l'enseignement supérieur et la recherche
. Ministère délégué à l'enseignement supérieur et à la recherche, 2006, disponible sur

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

http://www.breves-de-maths.fr/les-mathematiques-des-trous-noirs/

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/

# VIDEOS

Colloquium, Université de Nice, mars 2016 : "Questions de cosmologie mathématique"

# Prépublications / Preprints

• 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

Résumés / Abstracts

• A. BACHELOT. Problème inverse de diffusion non linéaire. C. R. Acad. Sc. Paris, Série I, 293:121-124, 1981.

We prove the Scattering Operator for the Non Linear Klein-Gordon Equation with interacting $q(x)u^3$ determines the coupling potential $q(x)$.

• A. BACHELOT , B. HANOUZET.Applications bilinéaires compatibles avec un système différentiel à coefficients variables. C. R. Acad. Sc. Paris, Série I, 299:543-546, 1984.

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.

• A. BACHELOT. Problème de Cauchy pour des systèmes hyperboliques semi-linéaires. Ann. Inst. Henri Poincaré, Analyse non linéaire, 1(6):453-478, 1984.

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.

• A. BACHELOT. Convergence dans $L^p(\BbbR^n+1)$ de la solution de l'équation de Klein-Gordon vers celle de l'équation des ondes. Ann. Fac. Sci. Toulouse, 8(1):37-60, 1986.

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 $$\square u+m\sp 2u=\sum\sb{k\ge 1}q\sb k(x)\vert u\vert\sp{2k}u.$$

• A. BACHELOT. Equipartition de l'énergie pour les systèmes hyperboliques et formes compatibles. Ann. Inst. Henri Poincaré - Physique théorique, 46(1):45-76, 1987.

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.

• A. BACHELOT, V. PETKOV.Existence des opérateurs d'ondes pour les systèmes hyperboliques avec un potentiel périodique en temps. Ann. Inst. Henri Poincaré - Physique théorique, 47(4):383-428, 1987.

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.

• A. BACHELOT. Problème de Cauchy global pour des systèmes de Dirac-Klein-Gordon. Ann. Inst. Henri Poincaré - Physique théorique, 48(4): 387-422, 1988.

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.

• A. BACHELOT. Global existence of large amplitude solutions for Dirac-Klein-Gordon systems in Minkowski space. In Non linear Hyperbolic Problems, volume 1402 of Lecture Notes in Math., pages 99-113. Springer Verlag, 1989.

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.

• A. BACHELOT. Global existence of large amplitude solutions for non linear massless Dirac equations. Portugaliae Math., 46(5):455-473, 1989.

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.

• A. BACHELOT. Gravitationnal Scattering of Electromagnetic Field by Schwarzschild Black Hole. Ann. Inst. Henri Poincaré - Physique théorique, 54(3):261-320, 1991.

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)].

• A. BACHELOT. Scattering of electromagnetic field by de Sitter-Schwarzschild black-hole. In Non linear hyperbolic equations and field theory, volume 253 of Research Notes in Mathematics, pages 23-35. Pitman, 1992.

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.

• A. BACHELOT, J-P. NICOLAS.Equation non linéaire de Klein-Gordon dans des métriques de type Schwarzschild. C. R. Acad. Sc. Paris, Série I, 316:1047-1050, 1993.

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.

• A. BACHELOT, A. MOTET-BACHELOT. Les résonances d'un trou noir de Schwarzschild. Ann. Inst. Henri Poincaré - Physique théorique, 59(1):3-68, 1993.

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.

• A. BACHELOT. Asymptotic Completeness for the Klein-Gordon Equation on the Schwarzschild Metric. Ann. Inst. Henri Poincaré - Physique théorique, 61(4):411-441, 1994.

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.

• A. BACHELOT, V. LANGE. Time dependent integral method for Maxwell's system with impedance boundary condition. In 10th International Conference on Boundary Element Technology, BETECH95, pages 137-144. Computational Mechanics Publications, 1995.

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.

• A. BACHELOT, V. LUBET. On the coupling of boundary element and finite element methods for a time problem. In G. Cohen, editor, Third International Conference on Mathematical and Numerical Aspects of Wave Propagation Phenomena, pages 130-139. SIAM-INRIA, 1995.

Abstract. We consider the problem of scattering transient waves, by an inhomogeneous object, in Èsup2(dfo()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.

• A. BACHELOT. Scattering of Scalar Fields by Spherical Gravitational Collapse. J. Math. Pures Appl., 76(2):155-210, 1997.

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.

• A. BACHELOT, A. PUJOLS. Décroissance de l'énergie locale d'une onde diffusée par un obstacle inhomogène. Rev. Roumaine Math. Pures et Appl., 41(7-8):451-459, 1997.

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.

• A. BACHELOT. Quantum Vacuum Polarization at the Black-Hole Horizon. Ann. Inst. Henri Poincaré - Physique théorique, 67(2):181-222 1997.

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.

• A. BACHELOT, P. CHARRIER, A. PUJOLS, D. ROUART. Parallel Algorithm for Solving Time Convolution Equations and Application to CEM. In 8th SIAM Conference on Parallel Processing for Scientific Computing, Minneapolis 1997, (CD-ROM). SIAM, 1997.

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.

• A. BACHELOT. The Hawking Effect. Ann. Inst. Henri Poincaré - Physique théorique, 1998.

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.

• A. BACHELOT. Creation of fermions at the charged black-hole horizon. Ann. Henri Poincaré, 2000.

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.

• A. BACHELOT, L. BOUNHOURE, A. PUJOLS. Couplage éléments finis-potentiels retardés pour la diffraction électromagnétique par un obstacle inhomogène. Numer. Math., 2001.

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.

• A. BACHELOT. Global properties of the wave equation on non globally hyperbolic manifolds. J. Math. Pures Appl., 2001.

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.

• A. BACHELOT. Superradiance and Scattering of the Charged Klein-Gordon Field by a Steplike Electrostatic Potential. J. Math. Pures Appl., 2004.

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.

• A. BACHELOT. Global Cauchy problem for semilinear hyperbolic systems with non-local nonlinearities. Applications to Dirac equations. J. Math. Pures Appl., 2006.

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.

•

• A. BACHELOT. Global waves with non positive energy in general relativity. Serdica. Math. J., 2008.

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].

•

• A. BACHELOT. The Dirac System on the Anti-De Sitter Universe. Comm. Math. Phys., 2008.

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.

• A. BACHELOT. Wave Propagation and Scattering for the RS2 Brane Cosmology Model. J. Hyperbolic Differ. Equ., 2009.

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.

• A. BACHELOT. The Klein-Gordon Equation in Anti-de Sitter Cosmology, J. Math. Pures Appl., 96 : 527–554, 2011.

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.

• A. BACHELOT. New Dynamics in the Anti-de Sitter Universe AdS^5, Comm. Math. Phys., 320, 723-759, 2013.

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.

• A. BACHELOT. On the Klein-Gordon equation near a De Sitter brane in an Anti-de Sitter bulk,  arXiv:1402.1071v3

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.

• A. BACHELOT. Waves in the Witten Bubble of Nothing and the Hawking Wormhole, arXiv:1601.03682
• We investigate the propagation of the scalar waves in the Witten space-time called "bubble of nothing" and in its remarkable sub-manifold, the lorentzian Hawking wormhole. Due to the global hyperbolicity, the global Cauchy problem is well-posed in the fonctional framework associated with the energy. We perform a complete spectral analysis that allows to get an explicit form of the solutions in terms of special functions. When the effective mass is non zero, the profile of the waves is asymptotically almost periodic in time. In opposite, the massless case is dispersive. We develop the scattering theory, classical as well as quantum. There is no creation of particle. The resonances can be defined in the massless case and they are purely imaginary.

• A. BACHELOT-MOTET, A. BACHELOT. Waves on accelerating dodecahedral universes, arXiv:1609.00806
• We investigate the wave propagation on a compact 3-manifold of constant positive curvature with a non trivial topology, the Poincar\'e dodecahedral space, when the scale factor is exponentially increasing. We prove the existence of a limit state as t tends to infinity and we get its analytic expression. The deep sky is described by this asymptotic profile thanks to the Sachs-Wolfe formula. We transform the Cauchy problem into a mixed problem posed on a fundamental domain determined by the quaternionic calculus. We perform an accurate scheme of computation: we employ a variational method using a space of second order finite elements that is invariant under the action of the binary icosahedral group.

• A. BACHELOT. Wave asymptotics at a cosmological time-singularity, arXiv:1806.01543
• We investigate the propagation of the scalar waves in the FLRW universes beginning with a Big Bang and ending with a Big Crunch, a Big Rip, a Big Brake or a Sudden Singularity. We obtain the sharp description of the asymptotics for the solutions of the linear Klein-Gordon equation, and similar results for the  semilinear equation with a subcritical exponent. We prove that the number of cosmological particle creations is finite under general assumptions on the initial Big Bang and the final Big Crunch or Big Brake.

Directions de thèses / Doctoral Students

1. Agnès PUJOLS, 31.10.1991, Ingénieur CEA.
2. Jean-Philippe NICOLAS, 10.01.1994, Habilitation : 26.02.1999, Professeur des universités, Université de Brest.
3. Valérie LUBET, 5.12.1994, Professeur des écoles.
4. Virginie LANGE, 28.10.1995, Ingénieur DASSAULT.
5. Agnès LECOMPTE, 7.07.1997, Ingénieur ASTRIUM.
6. Laurent BOUNHOURE, 4.12.1998, Ingénieur CS.
7. Eric DARRIGRAND, 25.09.2002, Maître de conférence, Université Rennes 1.
8. Fabrice MELNYK, 12.12.2002, Professeur second degrés.
9. Ludovic GATARD, 24.09.2007, Professeur agrégé.
10. Jacques MORICE, 17.10.2007, Ingénieur INRIA (CIRIC).

Retour aux Publications / Back to Publications

Alain BACHELOT

Distinguished Professor
University
of Bordeaux

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
Building A33, room 283
E-mail : alain.bachelot@u-bordeaux.fr

Main fields of interest
Hyperbolic Partial Differential Equation
Scattering Theory
Quantum Fields in General Relativity
Brane Cosmology
Black-Holes
Numerical Electromagnetism
More details :

Curriculum Vitae

Mathematical Genealogy

Publications

Videos

• Agrégation (1979)
• PhD thesis (1981)
• Habilitation (1988)

Distinctions :

Positions at the University of Bordeaux :

Main fields of research

• Linear and Non Linear Fields Equations on Minkowski Space-Time

• Inverse Scattering for Non Linear Klein-Gordon Equation (1981)
• Global Cauchy Problem with Non Regular Data for the Klein-Gordon-Schrödinger System (1983)
• Product of Vectorial Distributions in Besov Spaces (1983)
• Partition of Energy for Hyperbolic Systems(1985)
• Scattering by T-Periodic Potential for Hyperbolic Systems (with V. Petkov, 1987)
• Global Cauchy Problem for Lorentz-Invariant Dirac-Klein-Gordon Systems (1988)
• Global Solutions with Large Amplitude for Relativistic Dirac-Klein-Gordon Systems (1988-1989)
• Global Cauchy Problem for Semilinear Hyperbolic Systems with Non-Local Nonlinearities (2005)

• Black-Holes and Mathematical Cosmology (Main Field of Research)

• Maxwell System: Existence and Asymptotic Completeness of Wave Operators on Schwarzschild (1991) or De Sitter-Schwarzschild (1992) Metrics
• Regge-Wheeler Equation: Existence Of the Scattering Matrix S, Meromorphic Continuation of S, Computations of the Black-Hole Resonances on CRAY 2 (with A. Bachelot-Motet., 1993)
• Non Linear Klein-Gordon Equation on Schwarzschild metric : Global Cauchy Problem, Asymptotic Profiles (with J-P. Nicolas, 1993)
• Linear Klein-Gordon Equation on Schwarzschild metric : Asymptotic Completeness, Second Quantization, Quantum Scattering Operator (1994)
• Scattering of Klein-Gordon Field by Gravitational Collapse (1995)
• Quantum Vacuum Polarization at the Black-Hole Horizon (Hawking Radiation) (1997)
• Hawking Effect (1999)
• Radiation of Particles and Antiparticles by a Charged Black-Hole (1999)
• Causality violation (2001)
• Superradiance of the charged Black-Holes (2004)
• Global Dirac-Klein-Gordon Fields on Black-Holes Manifolds (2005)
• Global Dirac field on the Anti-De-Sitter Universe (2007)
• Waves and Scattering in the Randall-Sundrum Model of Brane Cosmology (2009)
• The Klein-Gordon equation in the Anti-de Sitter cosmology (2011)
• New boundary conditions at the time-like infinity of the Anti-de Sitter universe (2012)
• Linear stability of the De Sitter Brane (2014)
• Witten Bubble of Nothing, Hawking Lorentzian Wormhole (2016)
• Waves on accelerating dodecahedral universes (2017)
• Wave asymptotics at a cosmological time-singularity (2018)

• Computational Electromagnetism
• Time Dependent Integral Method for Maxwell System 3D+1 (with A. Pujols, 1991)
• Coupling Boundary Elements / Finite Elements Methods in 3D+1 (with V. Lubet, 1994)
• Integral Method for Absorbing Obstacle in 3D+1(with V. Lange 1995)
• Parallel Algorithm (with  P. Charrier, 1997)
• Scattering by Inhomogeneous Obstacles (with L. Bounhoure, 1998)
• Fast multipole method / Microlocal Discretization / High Order Finite Elements (with E. Darrigrand, K. Mer, 2000-2007)
• Fast multipole method for the radiosity (avec J. Morice, K. Mer, 2007)