Papers by Blaise TCHAPNDA
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2002
We prove the local-in-time existence of the solution of the Cauchy problem for the Maxwell equati... more We prove the local-in-time existence of the solution of the Cauchy problem for the Maxwell equations for an exact field, with current generated by a distribution function, subject to the Vlasov equation.
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...
Classical and Quantum Gravity, 2004
Results on the behaviour in the past time direction of cosmological models with collisionless mat... more Results on the behaviour in the past time direction of cosmological models with collisionless matter and a cosmological constant Λ are presented. It is shown that under the assumption of non-positive Λ and spherical or plane symmetry the area radius goes to zero at the initial singularity. Under a smallness assumption on the initial data, these properties hold in the case of hyperbolic symmetry and negative Λ as well as in the positive Λ case. Furthermore in the latter cases past global existence of spatially homogeneous solutions is proved for generic initial data. The early-time asymptotics is shown to be Kasner-like for small data.
Classical and Quantum Gravity, 2003
The behaviour of expanding cosmological models with collisionless matter and a positive cosmologi... more The behaviour of expanding cosmological models with collisionless matter and a positive cosmological constant is analysed. It is shown that under the assumption of plane or hyperbolic symmetry the area radius goes to infinity, the spacetimes are future geodesically complete, and the expansion becomes isotropic and exponential at late times. This proves a form of the cosmic no hair theorem in this class of spacetimes.
Mathematical Proceedings of the Cambridge Philosophical Society, 2005
The results on the local existence and continuation criteria obtained by G. Rein in are extended ... more The results on the local existence and continuation criteria obtained by G. Rein in are extended to the case with a non-zero cosmological constant. It is also shown that for the spherically symmetric case and a positive cosmological constant there is a large class of initial data with global existence and inflationary asymptotics in the future as in the case of plane or hyperbolic symmetry treated in . Furthermore we analyze the behaviour of the energy-momentum tensor at late times.
Annales Henri Poincaré, 2007
Some future global properties of cosmological solutions for the Einstein-Vlasov-Maxwell system wi... more Some future global properties of cosmological solutions for the Einstein-Vlasov-Maxwell system with surface symmetry are presented. Global existence is proved, the homogeneous spacetimes are future complete, and the same is true for inhomogeneous plane-symmetric solutions with small initial data. In the latter case some decay properties are also obtained at late times. Similar but slightly weaker results hold for hyperbolic symmetry.
Advances in Theoretical and Mathematical Physics, 2011
We consider plane-symmetric spacetimes satisfying Einstein's field equations with positive cosmol... more We consider plane-symmetric spacetimes satisfying Einstein's field equations with positive cosmological constant, when the matter is a fluid whose pressure is equal to its mass-energy density (i.e., a so-called stiff fluid). We study the initial-value problem for the associated Einstein equations and establish a global existence result. The late-time asymptotics of solutions is also rigorously derived, and we conclude that the spacetime approaches the de Sitter spacetime while the matter disperses asymptotically. A technical difficulty dealt with here lies in the fact that solutions may contain vacuum states as well as velocities approaching the speed of light, both possibilities leading to singular behavior in the evolution equations.
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.
Structure of solutions near the initial singularity
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 difficulties which arise in that case are overcome by using the fact that the dynamical system of interest is of geometrical origin and thus has useful invariant manifolds.
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.
Some future global properties of cosmological solutions for the Einstein-Vlasov-Maxwell system wi... more Some future global properties of cosmological solutions for the Einstein-Vlasov-Maxwell system with surface symmetry are presented. Global existence is proved, the homogeneous spacetimes are future complete for causal trajectories, and the same is true for inhomogeneous plane-symmetric solutions with small initial data. In the latter case some decay properties are also obtained at late times. Similar but slightly weaker results hold for hyperbolic symmetry.
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.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012
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.
Journal of Hyperbolic Differential Equations, Sep 1, 2008
The Einstein equations with a positive cosmological constant are coupled to the pressureless perf... more The Einstein equations with a positive cosmological constant are coupled to the pressureless perfect fluid matter in plane symmetry. Under suitable restrictions on the initial data, the resulting Einstein-dust system is proved to have a global classical solution in the future time direction. Some late time asymptotic properties are obtained as well.
Proceedings of the Royal Society of London a Mathematical Physical and Engineering Sciences, Aug 8, 2002
We prove the local-in-time existence of the solution of the Cauchy problem for the Maxwell equati... more We prove the local-in-time existence of the solution of the Cauchy problem for the Maxwell equations for an exact field, with current generated by a distribution function, subject to the Vlasov equation.
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Papers by Blaise TCHAPNDA