The Global Analysis of Fluids in General Relativity

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The focus of this proposal is the study of the global nonlinear behavior of solutions to the relativistic Euler equations, which are the fundamental PDEs of relativistic fluid mechanics. In particular, the principal investigator will study the effect that spacetime expansion has on the global behavior of fluids. A primary goal is to understand the minimal expansion rate that is capable of suppressing the formation of shocks in solutions that are launched by small, smooth initial data. A related secondary goal is to find conditions on smooth initial data that guarantee the evolutionary formation of shocks. In a broader context, this work will advance our understanding of the global behavior of solutions to quasilinear hyperbolic PDEs on expanding Lorentzian manifolds. Attacking these problems requires the development and application of mathematical techniques lying at the interface of geometry, analysis, and fluid mechanics. Specifically, the principal investigator will use geometric energy methods to derive dissipative and dispersive estimates for solutions to hyperbolic PDEs.The projects in this proposal are expected to help provide rigorous mathematical justification for some of the most fundamental predictions of cosmology, many of which are based on the behavior of explicit solutions to the relativistic Euler equations; in cosmology, much of the "normal matter" content of the universe is assumed to be effectively modeled by a fluid. It is especially important to understand the effect that spacetime expansion has on the behavior of fluids, for experimental observations indicate that our own universe is undergoing accelerated expansion. Although the physical picture of the fluid behavior set forth by the standard cosmological model is by now well-established, a full mathematical justification of its predictions has yet to emerge. In particular, in order for the explicit fluid solutions to have predictive value, it is essential to show that they are globally stable under small perturbations of their initial conditions; this is one of the major proposal goals. On the other hand, the principal investigator's investigation of shock formation may expose some limitations of the fluid model in cosmology. In addition to conducting research, the principal investigator will also supervise undergraduate research projects connected to the main themes of this proposal.