Many of our most celebrated physical theories are based on wave-like partial differential equations (PDEs). Important examples include the Einstein equations of general relativity (which form the basis of modern cosmology), the Euler equations of fluid mechanics, the equations of elasticity, and the equations of crystal optics. Despite hundreds of years of mathematical progress, major gaps remain in our understanding of solutions. For example, although it is expected that solutions often "become infinite" there are few cases in which mathematicians have found a proof. Moreover, in other contexts, it is not even known if the equations have solutions. Thus, we still do not have a definitive understanding of the physical predictions of many equations of classical physics. The main obstacle is that the aforementioned equations are extremely difficult to study except under unrealistic simplifying assumptions. However, there have been some recent advancements in resolving some of these difficulties. For example, recent results provide a detailed description of Big Bang formation in solutions to the Einstein equations and shocks in solutions to the Euler and related equations. A major goal of this project is to extend these results to apply to other physically relevant equations and regimes. This work will provide new insight on the long-time behavior of waves and on the ways in which various physical theories can break down.
This research project involves the rigorous study of quasilinear hyperbolic PDEs and has three main research goals. The first is to provide a detailed description of singularity formation in various quasilinear wave-like equations of physical significance, with an emphasis on avoiding symmetry assumptions whenever possible. Examples include various Einstein-matter equations, the Euler equations, and the equations of elasticity. This research will expand our understanding of singularity formation by enlarging the class of equations and the class of initial conditions that are known to lead to blow-up. The second research goal is to prove local well-posedness for a class of physically relevant quasilinear hyperbolic problems with a free boundary along which the hyperbolicity of the equations degenerates. The third is to develop new tools for understanding the behavior of solutions to quasilinear hyperbolic PDEs with multiple characteristics in contexts where knowledge of the precise characteristics is essential. An ultimate goal is to use these tools to prove that for such equations, under suitable assumptions on the nonlinearities and data, shock formation occurs and is stable. This proposal has several educational components that are intimately connected to the research problems. Many of the problems have components that can be investigated by advanced undergraduates, and the PI plans to supervise undergraduate research projects. There are also components suitable for PhD students. Moreover, the PI is organizing, with three co-organizers, a pair of summer schools targeted at advanced undergraduates and beginning graduate students from across the US and beyond. The PI will also develop curriculum materials that will be made freely available to the public online.
