On the questions of local and global well-posedness for the hyperbolic PDEs occurring in some relativistic theories of gravity and electromagnetism

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.Ph.D.Includes bibliographical references (p. 140-143)