CAREER: Geometric Methods in Hyperbolic Partial Differential Equations

我们许多最著名的物理理论都是建立在波状偏微分方程（PDEs）的基础上的。重要的例子包括广义相对论的爱因斯坦方程（它构成了现代宇宙学的基础）、流体力学的欧拉方程、弹性方程和晶体光学方程。尽管数学有了数百年的进步，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“变得无限”，但数学家们却很少找到证明。此外，在其他情况下，甚至不知道方程是否有解。因此，我们对许多经典物理方程的物理预测仍然没有一个明确的理解。主要的障碍是，除非在不切实际的简化假设下，否则上述方程极其难以研究。不过，最近在解决其中一些困难方面取得了一些进展。例如，最近的结果在爱因斯坦方程的解中提供了大爆炸形成的详细描述，在欧拉方程和相关方程的解中提供了冲击的详细描述。该项目的一个主要目标是将这些结果扩展到其他物理相关方程和体系中。这项工作将为波的长期行为和各种物理理论可能崩溃的方式提供新的见解。本研究项目涉及拟线性双曲偏微分方程的严格研究，主要有三个研究目标。首先是在各种具有物理意义的拟线性波状方程中提供奇点形成的详细描述，重点是尽可能避免对称假设。例子包括各种爱因斯坦-物质方程、欧拉方程和弹性方程。这项研究将通过扩大已知导致爆炸的方程和初始条件的类别来扩展我们对奇点形成的理解。第二个研究目标是证明一类具有自由边界的拟线性双曲型问题的局部适定性。第三是开发新的工具来理解具有多个特征的拟线性双曲偏微分方程的解的行为，在这种情况下，精确特征的知识是必不可少的。最终目标是使用这些工具来证明，对于这样的方程，在对非线性和数据的适当假设下，激波形成发生并且是稳定的。这个建议有几个与研究问题密切相关的教育组成部分。许多问题的组成部分可以由高年级本科生进行研究，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正在与三家联合组织者组织两所暑期学校，目标是来自美国各地和其他地区的高级本科生和初级研究生。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.