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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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Text
TitleInfo (ID = T-1)
Title
On the questions of local and global well-posedness for the hyperbolic PDEs occurring in some relativistic theories of gravity and electromagnetism
Identifier (type = hdl)
http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.17393
Identifier
ETD_999
Language
LanguageTerm
English
Genre (authority = marcgt)
theses
Subject (ID = SBJ-1); (authority = RUETD)
Topic
Mathematics
Subject (ID = SBJ-2); (authority = ETD-LCSH)
Topic
Differential equations, Partial
Subject (ID = SBJ-3); (authority = ETD-LCSH)
Topic
Differential equations, Hyperbolic
Subject (ID = SBJ-4); (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.
PhysicalDescription
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xi, 144 pages
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Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references (p. 140-143).
Name (ID = NAME-1); (type = personal)
NamePart (type = family)
Speck
NamePart (type = given)
Jared R.
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author
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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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Advisory Committee
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Michael K.-H. Kiessling
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Tahvildar-Zadeh
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A.
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co-chair
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Advisory Committee
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A. Shadi Tahvildar-Zadeh
Name (ID = NAME-4); (type = personal)
NamePart (type = family)
Goodman
NamePart (type = given)
Roe
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internal member
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Advisory Committee
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Roe Goodman
Name (ID = NAME-5); (type = personal)
NamePart (type = family)
Klainerman
NamePart (type = given)
Sergiu
Role
RoleTerm (authority = RULIB)
outside member
Affiliation
Advisory Committee
DisplayForm
Sergiu Klainerman
Name (ID = NAME-1); (type = corporate)
NamePart
Rutgers University
Role
RoleTerm (authority = RULIB)
degree grantor
Name (ID = NAME-2); (type = corporate)
NamePart
Graduate School - New Brunswick
Role
RoleTerm (authority = RULIB)
school
OriginInfo
DateCreated (qualifier = exact)
2008
DateOther (qualifier = exact); (type = degree)
2008-05
Location
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NjNbRU
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TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier (type = local)
rucore19991600001
Identifier (type = doi)
doi:10.7282/T3TX3FQS
Genre (authority = ExL-Esploro)
ETD doctoral
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The author owns the copyright to this work.
Copyright
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Availability
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Open
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Name
Jared Speck
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Copyright holder
Affiliation
Rutgers University. Graduate School - New Brunswick
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Non-exclusive ETD license
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Author Agreement License
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