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Future stability of the Einstein-Maxwell-Scalar field system and non-linear wave equations coupled to generalized massive-massless Vlasov equations
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of two articles related to mathematical relativity theory.

In the first article we prove future stability of certain spatially homogeneous solutionsto Einstein’s field equations. The matter model is assumed to consist of an electromagnetic field and a scalar field with a potential creating an accelerated expansion. Beside this, more general properties concerning Einstein’s field equation coupled to a scalar field and an electromagnetic field are settled. The most important of these questions are the existence of a maximal globally hyperbolic development and the Cauchy stability of solutions to the initial value problem.

In the second article we consider Einstein’s field equations where the matter model consists of two momentum distribution functions. The first momentum distribution function represents massive matter, for instance galactic dust, and the second represents massless matter, for instance radiation. Furthermore, we require that each of the momentum distribution functions shall satisfy the Vlasov equation. This means that the momentum distribution functions represent collisionless matter. If Einstein’s field equations with such a matter model is expressed in coordinates and if certain gauges are fixed we get a system of integro-partial differential equations we shall call non-linear wave equations coupled to generalized massive-massless Vlasov equations. In the second article we prove that the initial value problem associated to this kind of equations has a unique local solution.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2012. , vi, 19 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 12:03
National Category
Mathematical Analysis Geometry
Identifiers
URN: urn:nbn:se:kth:diva-93891ISBN: 978-91-7501-294-0 (print)OAI: oai:DiVA.org:kth-93891DiVA: diva2:524534
Public defence
2012-05-21, Sal F3, Lindstedtsvägen 26, KTH, 10:00 (English)
Opponent
Supervisors
Note
QC 20120503Available from: 2012-05-03 Created: 2012-05-02 Last updated: 2012-05-03Bibliographically approved
List of papers
1. Future Stability of the Einstein-Maxwell-Scalar Field System
Open this publication in new window or tab >>Future Stability of the Einstein-Maxwell-Scalar Field System
2011 (English)In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 12, no 5, 849-917 p.Article in journal (Refereed) Published
Abstract [en]

Ringstrom managed (in Invent Math 173(1):123-208, 2008) to prove future stability of solutions to Einstein's field equations when matter consists of a scalar field with a potential creating an accelerated expansion. This was done for a quite wide class of spatially homogeneous space-times. The methods he used should be applicable also when other kinds of matter fields are added to the stress-energy tensor. This article addresses the question whether we can obtain stability results similar to those Ringstrom obtained if we add an electromagnetic field to the matter content. Before this question can be addressed, more general properties concerning Einstein's field equation coupled to a scalar field and an electromagnetic field have to be settled. The most important of these questions are the existence of a maximal globally hyperbolic development and the Cauchy stability of solutions to the initial value problem.

National Category
Physical Sciences Other Physics Topics
Identifiers
urn:nbn:se:kth:diva-34400 (URN)10.1007/s00023-011-0099-y (DOI)000290668500002 ()2-s2.0-79956120392 (Scopus ID)
Note
QC 20110607Available from: 2011-06-07 Created: 2011-06-07 Last updated: 2017-12-11Bibliographically approved
2. Non-linear wave equations coupled to generalized massive-massless Vlasov equations
Open this publication in new window or tab >>Non-linear wave equations coupled to generalized massive-massless Vlasov equations
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Consider Einstein’s field equations where the matter model consists oftwo momentum distribution functions. Let the first momentum distribution function represent massive matter, for instance galactic dust, and let the second represent massless matter, for instance radiation. Furthermore,let us require that each of the momentum distribution functions shall satisfy the Vlasov equation. This means that the momentum distribution functions represent collisionless matter. If Einstein’s field equations withsuch a matter model is expressed in coordinates and if certain gauges arefixed we get a system of integro-partial differential equations we shall call non-linear wave equations coupled to generalized massive-massless Vlasovequations. We prove that the initial value problem associated to this kindof equations has a unique local solution. Moreover, we prove a continuation criterion for the solution.

Keyword
Einstein's field equations, Vlasov equations
National Category
Geometry Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-93786 (URN)
Available from: 2012-05-02 Created: 2012-04-26 Last updated: 2012-05-03Bibliographically approved

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