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

Abstract

OF THE DISSERTATION On the Questions of Local and Global Well-Posedness for the Hyperbolic PDEs Occurring in Some Relativistic Theories of Gravity and Electromagnetism by Jared R. Speck Dissertation Directors: Michael K.-H. Kiessling and A. Shadi Tahvildar-Zadeh 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: • We give a sharp non-local criterion for the formation of singularities in planesymmetric 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 planesymmetric 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.