| The two hyperbolic systems of PDEs we consider in this work are the source-free Maxwell-Born-Infeld (MBI) field equations and the Euler-Nordstrom 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-Nordstrom system with a cosmological constant (ENkappa) for initial data that are an HN perturbation (not necessarily small) of a uniform, quiet fluid, for N ≥ 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 HN, namely symmetric hyperbolicity and strict hyperbolicity, do not apply to the ENkappa system, while Christodoulou's techniques apply to all hyperbolic systems derivable from a Lagrangian, of which the ENkappa system is an example. (3) We insert the speed of light c as a parameter into the ENkappa system (and designate the family of systems ENck ) 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 (EPkappa). Using energy currents, we prove that for fixed initial data, as c goes to infinity, the solutions to the ENck system converge uniformly on a spacetime slab [0,T] x R3 to the solution of the EPkappa system. |
