The research under this award investigates connections between geometry and the mathematical theory of shock waves. This includes a continuation of the authors' recent work on developing a mathematical theory of shock-waves that applies to the Einstein equations - the equations that describe the time evolution of spacetime curvature, the gravitational field according to general relativity. Shock waves are relevent here because the compressible Euler equations appear as a subsystem of the Einstein equations, and emerge exactly in the limit of low velocities and weak gravitational fields. The work includes the construction of a new cosmological model in which the expansion of the universe, (as measured by the Hubble constant), is modeled as the effect of a great explosion that generates a shock wave at the leading edge, something like a nuclear explosion into a static background, except on an enormous scale, instead of by the standard cosmological model in which it is assumed that the entire universe is expanding at a rate measured by the Hubble law. If successful, the results will provide the first and simplest model for cosmology under the assumption that the Hubble constant measures only a localized expansion, and consistent with the energy density and background radiation levels observed in the universe today. This research also has an interesting philosophical implication in that, if there were a shock wave present from the Big Bang, then, in contrast to the standard model of cosmology, there would be an associated loss of information that would make it impossible to reconstruct the details of the initial event from present data. In addition, a covariant version of Glimm's Method will be developed in an attempt to provide the first general existence theory of shock wave interactions for the Einstein equations. This is a step toward developing principles that can be applied to the numerical simulation of shock waves in general relativity. These studies will also have implications for classical fluids because the Einstein equations provide a unique setting in which natural geometrical constructs put a handle on the fluid dynamics. This work dovetails with ongoing research into how the large time behavior of shock wave solutions is related to the geometry of the Lie Algebra that gives the scattering picture for shock wave interactions. Thus this research connects the theory of shock waves and geometry on several levels.
Einstein's equations of general relativity describe the large scale development of the universe. Although they were first proposed more than eighty years ago, they still pose formidable mathematical challenges. This research will continue earlier work in which exact shock wave solutions of these equations were found. A mathematical theory will be developed that leads to alternatives of the standard "Big Bang" theory and that allows a general theoretical treatment of shock waves within the general theory of relativity. Since these equations contain the classical fluid dynamics equations as a special case, the research will also lead to new insights for these more familiar problems.
