CAREER: Geometric Methods in Hyperbolic Partial Differential Equations

我们许多最著名的物理理论都是基于波动偏微分方程（PDE）。重要的例子包括广义相对论的爱因斯坦方程（构成现代宇宙学的基础），流体力学的欧拉方程，弹性方程和晶体光学方程。尽管数学已经发展了数百年，但我们对解的理解仍然存在重大差距。例如，尽管人们期望解经常“变得无穷大”，但数学家找到证明的情况很少。此外，在其他情况下，甚至不知道方程是否有解。因此，我们仍然没有对许多经典物理方程的物理预测有明确的理解。主要的障碍是，上述方程极难研究，除非在不切实际的简化假设下。然而，最近在解决其中一些困难方面取得了一些进展。例如，最近的结果提供了一个详细的描述大爆炸形成的解决方案，爱因斯坦方程和冲击的解决方案，欧拉方程和相关的。这个项目的一个主要目标是将这些结果扩展到其他物理相关的方程和制度。这项工作将为波的长期行为以及各种物理理论的崩溃提供新的见解。本研究计画主要针对拟线性双曲型偏微分方程进行严谨的研究，并有三个主要的研究目标。第一个是提供一个详细的描述奇异性的形成在各种准线性波动方程的物理意义，重点是避免对称性假设，只要可能。例子包括各种爱因斯坦物质方程，欧拉方程和弹性方程。这项研究将扩大我们的理解奇异性的形成，扩大类方程和类的初始条件，已知导致爆破。第二个研究目标是证明一类具有自由边界的拟线性双曲问题的局部适定性，其中方程的双曲性沿自由边界沿着退化。第三是开发新的工具，了解的行为的解决方案，拟线性双曲型偏微分方程的多个特点的背景下，知识的精确特性是必不可少的。最终的目标是使用这些工具来证明，对于这样的方程，在适当的假设下的非线性和数据，冲击形成的发生和稳定。 这项建议有几个教育组成部分，与研究问题密切相关。许多问题都有可以由高年级本科生调查的组成部分，PI计划监督本科生的研究项目。也有适合博士生的组件。此外，PI正在与三个联合组织者一起组织一对针对来自美国及其他地区的高级本科生和研究生的暑期学校。PI还将开发课程材料，并在网上免费提供给公众。

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.