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Papers by Blaise TCHAPNDA

Research paper thumbnail of The Maxwell--Vlasov system for an exact field on a curved space-time

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2002

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Research paper thumbnail of Oscillatory singularities in Bianchi models with magnetic fields

An idea which has been around in general relativity for more than forty years is that in the appr... more An idea which has been around in general relativity for more than forty years is that in the approach to a big bang singularity solutions of the Einstein equations can be approximated by the Kasner map, which describes a succession of Kasner epochs. This is already a highly non-trivial statement in the spatially homogeneous case. There the Einstein equations reduce to ordinary differential equations and it becomes a statement that the solutions of the Einstein equations can be approximated by heteroclinic chains of the corresponding dynamical system. For a long time progress on proving a statement of this kind rigorously was very slow but recently there has been new progress in this area, particularly in the case of the vacuum Einstein equations. In this paper we generalize some of these results to the Einstein-Maxwell equations. It turns out that this requires new techniques since certain eigenvalues are in a less favourable configuration in the case with a magnetic field. The diff...

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Research paper thumbnail of Structure of solutions near the initial singularity for the surface-symmetric Einstein–Vlasov system

Classical and Quantum Gravity, 2004

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Research paper thumbnail of Global existence and asymptotic behaviour in the future for the Einstein--Vlasov system with positive cosmological constant

Classical and Quantum Gravity, 2003

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Research paper thumbnail of The surface-symmetric Einstein–Vlasov system with cosmological constant

Mathematical Proceedings of the Cambridge Philosophical Society, 2005

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Research paper thumbnail of On Surface-Symmetric Spacetimes with Collisionless and Charged Matter

Annales Henri Poincaré, 2007

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Research paper thumbnail of Plane-symmetric spacetimes with positive cosmological constant: The case of stiff fluids

Advances in Theoretical and Mathematical Physics, 2011

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Research paper thumbnail of On the spherically symmetric Einstein-Yang-Mills-Higgs equations in Bondi coordinates

We revisit and generalize, to the Einstein-Yang-Mills-Higgs system, previous results of D. Christ... more We revisit and generalize, to the Einstein-Yang-Mills-Higgs system, previous results of D. Christodoulou and D. Chae concerning global solutions for the Einstein-scalar field and the Einstein-Maxwell-Higgs equations. The novelty of the present work is twofold. For one thing the assumption on the self-interaction potential is improved. For another thing explanation is furnished why the solutions obtained here and those proved by Chae for the Einstein-Maxwell-Higgs decay more slowly than those established by Christodoulou in the case of self-gravitating scalar fields. Actually this latter phenomenon stems from the non-vanishing local charge in Einstein-Maxwell-Higgs and Einstein-Yang-Mills-Higgs models.

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Research paper thumbnail of For the Surface-Symmetric

Structure of solutions near the initial singularity

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Research paper thumbnail of Oscillatory singularities in Bianchi models with magnetic fields

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Research paper thumbnail of On the Spherically Symmetric Einstein-Yang-Mills-Higgs Equations in Bondi Coordinates

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Research paper thumbnail of The Einstein-Vlasov system with cosmological constant in a surface-symmetric cosmological model: local existence and continuation criteria

The Einstein-Vlasov system describes a self-gravitating, collision-less gas within the framework ... more The Einstein-Vlasov system describes a self-gravitating, collision-less gas within the framework of general relativity. We investigate the initial value problem in a cosmological setting with surface symmetry and a non-zero cosmological constant and prove local existence and continuation criteria in both time directions. The continuation criterion says that as long as the maximum velocity remains bounded and the lapse function remains bounded then the solution can be continued. This applies to either time direction.

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Research paper thumbnail of And Charged Matter

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Research paper thumbnail of Yang-Mills System on a Curved Spacetime

We prove a local in time existence theorem for the Cauchy problem for the Yang-Mills system in te... more We prove a local in time existence theorem for the Cauchy problem for the Yang-Mills system in temporal gauge, with current generated both by a distribution function that satisfies the Vlasov equation, and an unknown Yang-Mills charge density, subject to a conservation equation. We prove a global in time existence theorem in the case of massless particles.

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Research paper thumbnail of On the spherically symmetric Einstein-Yang-Mills-Higgs equations in Bondi coordinates

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012

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Research paper thumbnail of The Plane Symmetric Einstein-Dust System with Positive Cosmological Constant

Journal of Hyperbolic Differential Equations, Sep 1, 2008

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Research paper thumbnail of The Maxwell-Vlasov system for an exact field on a curved space-time

Proceedings of the Royal Society of London a Mathematical Physical and Engineering Sciences, Aug 8, 2002

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Research paper thumbnail of The Maxwell--Vlasov system for an exact field on a curved space-time

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2002

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Research paper thumbnail of Oscillatory singularities in Bianchi models with magnetic fields

An idea which has been around in general relativity for more than forty years is that in the appr... more An idea which has been around in general relativity for more than forty years is that in the approach to a big bang singularity solutions of the Einstein equations can be approximated by the Kasner map, which describes a succession of Kasner epochs. This is already a highly non-trivial statement in the spatially homogeneous case. There the Einstein equations reduce to ordinary differential equations and it becomes a statement that the solutions of the Einstein equations can be approximated by heteroclinic chains of the corresponding dynamical system. For a long time progress on proving a statement of this kind rigorously was very slow but recently there has been new progress in this area, particularly in the case of the vacuum Einstein equations. In this paper we generalize some of these results to the Einstein-Maxwell equations. It turns out that this requires new techniques since certain eigenvalues are in a less favourable configuration in the case with a magnetic field. The diff...

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Research paper thumbnail of Structure of solutions near the initial singularity for the surface-symmetric Einstein–Vlasov system

Classical and Quantum Gravity, 2004

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Research paper thumbnail of Global existence and asymptotic behaviour in the future for the Einstein--Vlasov system with positive cosmological constant

Classical and Quantum Gravity, 2003

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Research paper thumbnail of The surface-symmetric Einstein–Vlasov system with cosmological constant

Mathematical Proceedings of the Cambridge Philosophical Society, 2005

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Research paper thumbnail of On Surface-Symmetric Spacetimes with Collisionless and Charged Matter

Annales Henri Poincaré, 2007

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Research paper thumbnail of Plane-symmetric spacetimes with positive cosmological constant: The case of stiff fluids

Advances in Theoretical and Mathematical Physics, 2011

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Research paper thumbnail of On the spherically symmetric Einstein-Yang-Mills-Higgs equations in Bondi coordinates

We revisit and generalize, to the Einstein-Yang-Mills-Higgs system, previous results of D. Christ... more We revisit and generalize, to the Einstein-Yang-Mills-Higgs system, previous results of D. Christodoulou and D. Chae concerning global solutions for the Einstein-scalar field and the Einstein-Maxwell-Higgs equations. The novelty of the present work is twofold. For one thing the assumption on the self-interaction potential is improved. For another thing explanation is furnished why the solutions obtained here and those proved by Chae for the Einstein-Maxwell-Higgs decay more slowly than those established by Christodoulou in the case of self-gravitating scalar fields. Actually this latter phenomenon stems from the non-vanishing local charge in Einstein-Maxwell-Higgs and Einstein-Yang-Mills-Higgs models.

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Research paper thumbnail of For the Surface-Symmetric

Structure of solutions near the initial singularity

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Oscillatory singularities in Bianchi models with magnetic fields

Bookmarks Related papers MentionsView impact

Research paper thumbnail of On the Spherically Symmetric Einstein-Yang-Mills-Higgs Equations in Bondi Coordinates

Bookmarks Related papers MentionsView impact

Research paper thumbnail of The Einstein-Vlasov system with cosmological constant in a surface-symmetric cosmological model: local existence and continuation criteria

The Einstein-Vlasov system describes a self-gravitating, collision-less gas within the framework ... more The Einstein-Vlasov system describes a self-gravitating, collision-less gas within the framework of general relativity. We investigate the initial value problem in a cosmological setting with surface symmetry and a non-zero cosmological constant and prove local existence and continuation criteria in both time directions. The continuation criterion says that as long as the maximum velocity remains bounded and the lapse function remains bounded then the solution can be continued. This applies to either time direction.

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Research paper thumbnail of And Charged Matter

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Yang-Mills System on a Curved Spacetime

We prove a local in time existence theorem for the Cauchy problem for the Yang-Mills system in te... more We prove a local in time existence theorem for the Cauchy problem for the Yang-Mills system in temporal gauge, with current generated both by a distribution function that satisfies the Vlasov equation, and an unknown Yang-Mills charge density, subject to a conservation equation. We prove a global in time existence theorem in the case of massless particles.

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Research paper thumbnail of On the spherically symmetric Einstein-Yang-Mills-Higgs equations in Bondi coordinates

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012

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Research paper thumbnail of The Plane Symmetric Einstein-Dust System with Positive Cosmological Constant

Journal of Hyperbolic Differential Equations, Sep 1, 2008

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Research paper thumbnail of The Maxwell-Vlasov system for an exact field on a curved space-time

Proceedings of the Royal Society of London a Mathematical Physical and Engineering Sciences, Aug 8, 2002

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