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On the questions of local and global well-posedness for the hyperbolic PDEs occurring in some relativistic theories of gravity and electromagnetism
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Title
On the questions of local and global well-posedness for the hyperbolic PDEs occurring in some relativistic theories of gravity and electromagnetism
Identifier
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http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.17393
Identifier
ETD_999
Language
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English
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theses
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Mathematics
Subject
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(authority = ETD-LCSH)
Topic
Differential equations, Partial
Subject
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(authority = ETD-LCSH)
Topic
Differential equations, Hyperbolic
Subject
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(authority = ETD-LCSH)
Topic
Gravitation
Subject
(ID = SBJ-5);
(authority = ETD-LCSH)
Topic
Electromagnetic theory
Abstract
The two hyperbolic systems of PDEs we consider in this work are the source-free Maxwell-Born-Infeld (MBI) field equations and the Euler-Nordstr??m system for gravitationally self-interacting fluids. The former system plays a central role in Kiessling's recently proposed self-consistent model of classical
electrodynamics with point charges, a model that does not suffer from the infinities found in the classical Maxwell-Maxwell model with point charges. The latter system is a scalar gravity caricature of the incredibly more complex Euler-Einstein system. The primary original contributions of the thesis can be summarized as follows:
1) We give a sharp non-local criterion for the formation of singularities in plane-symmetric solutions to the source-free MBI field equations. We also use a domain of dependence argument to show that 3-d initial data agreeing with certain plane-symmetric data on a large enough ball lead to solutions that form singularities in finite time. This work is an extension of a theorem of Brenier, who studied singularity formation in periodic plane-symmetric solutions.
2) We prove well-posedness for the Euler-Nordstr??m system with a cosmological constant k (EN_k) for initial data that are an H^N perturbation (not necessarily small) of a uniform, quiet fluid, for N [greater than]= 3. The method of proof relies on the framework of energy currents that has been recently developed by Christodoulou. We turn to this method out of necessity: two common frameworks for showing well-posedness in H^N, namely symmetric hyperbolicity and strict hyperbolicity, do not apply to the EN_k system, while Christodoulou's techniques apply to all hyperbolic systems derivable from a Lagrangian, of which the EN_k system is an example.
3) We insert the speed of light c as a parameter into the EN_k system (and designate the family of systems EN_k^c) in order to study the non-relativistic limit c to infinity. Taking the formal limit in the equations gives the Euler-Poisson system with a cosmological constant (EP_k). Using energy currents, we prove that for fixed initial data, as c goes to infinity, the solutions to the EN_k^c system converge uniformly on a spacetime slab [0,T] x R^3 to the solution of the EP_k system.
electrodynamics with point charges, a model that does not suffer from the infinities found in the classical Maxwell-Maxwell model with point charges. The latter system is a scalar gravity caricature of the incredibly more complex Euler-Einstein system. The primary original contributions of the thesis can be summarized as follows:
1) We give a sharp non-local criterion for the formation of singularities in plane-symmetric solutions to the source-free MBI field equations. We also use a domain of dependence argument to show that 3-d initial data agreeing with certain plane-symmetric data on a large enough ball lead to solutions that form singularities in finite time. This work is an extension of a theorem of Brenier, who studied singularity formation in periodic plane-symmetric solutions.
2) We prove well-posedness for the Euler-Nordstr??m system with a cosmological constant k (EN_k) for initial data that are an H^N perturbation (not necessarily small) of a uniform, quiet fluid, for N [greater than]= 3. The method of proof relies on the framework of energy currents that has been recently developed by Christodoulou. We turn to this method out of necessity: two common frameworks for showing well-posedness in H^N, namely symmetric hyperbolicity and strict hyperbolicity, do not apply to the EN_k system, while Christodoulou's techniques apply to all hyperbolic systems derivable from a Lagrangian, of which the EN_k system is an example.
3) We insert the speed of light c as a parameter into the EN_k system (and designate the family of systems EN_k^c) in order to study the non-relativistic limit c to infinity. Taking the formal limit in the equations gives the Euler-Poisson system with a cosmological constant (EP_k). Using energy currents, we prove that for fixed initial data, as c goes to infinity, the solutions to the EN_k^c system converge uniformly on a spacetime slab [0,T] x R^3 to the solution of the EP_k system.
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xi, 144 pages
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Ph.D.
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Includes bibliographical references (p. 140-143).
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Speck
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Jared R.
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Jared R. Speck
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Kiessling
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Michael
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chair
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Michael K.-H. Kiessling
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Tahvildar-Zadeh
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co-chair
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A. Shadi Tahvildar-Zadeh
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Goodman
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Roe
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Roe Goodman
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Klainerman
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Sergiu
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outside member
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Advisory Committee
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Sergiu Klainerman
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Rutgers University
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Graduate School - New Brunswick
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2008
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2008-05
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Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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doi:10.7282/T3TX3FQS
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ETD doctoral
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Jared Speck
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Rutgers University. Graduate School - New Brunswick
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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