On the Behavior of Solutions of Einstein's Equations and Other Geometric Nonlinear Partial Differential Equation Systems

点击翻译按钮获取中文摘要

A good portion of this award is devoted to the study of the formation of singularities, both in solutions of Einstein's equations and in solutions of Ricci flow. For Einstein's equations, one of the key issues is Strong Cosmic Censorship ("SCC"): Is it true that the ubiquitous geodesic incompleteness predicted by the Hawking-Penrose theorems generically involve spacetime curvature (and therefore gravitational tidal) blowup? At least in certain families of spacetimes defined by isometries, the determination that the solutions exhibit "AVTD" behavior (dominance of time derivatives of the fields over space derivatives near the singularity) is a very helpful tool for studying SCC, and is interesting in its own right. Some of our work involves the development of a framework for proving that there are large sets of smooth (non-analytic) solutions in these families which show AVTD behavior. Our studies of singularities in Ricci flow have focused on those known as "neckpinches". We have, in particular, shown that there is a discretely parametrized family of Ricci flow solutions which develop "degenerate neckpinches", with the geometry asymptotically approaching a "Bryant soliton" at a prescribed rate (consistent with Type 2 behavior). These solutions are all rotationally symmetric. Wanting to know if the detailed asymptotic behavior seen in these solutions is also characteristic of non-rotationally symmetric solutions, we have chosen to first address this issue in the case of mean curvature flow ("MCF"), since MCF neckpinches share many features with those of Ricci flow, and since it is easier with MCF to compare non-rotationally symmetric solutions with rotationally symmetric ones.In the projects discussed above, as well as in many others in my research program, the primary goal is to develop an understanding of the general behavior of large classes of solutions of complicated nonlinear systems of partial differential equations. Such equation systems, which play a major role in our modeling of physical phenomena at the terrestrial scale as well as at the astrophysical/cosmological scale, cannot be solved explicitly. However, using techniques such as those developed and used in this proposal, one can learn a tremendous amount of practical information about the behavior of solutions of these models, and therefore make very useful predictions regarding the behavior of the physical systems modeled by these equation systems.