CAREER: Geometric Methods in Hyperbolic Partial Differential Equations

职业：双曲偏微分方程中的几何方法

• 批准号: 1914537
• 负责人: Jared Speck
• 金额: $ 14.09万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-09-01 至 2022-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1914537&HistoricalAwards=false
• 关键词: CAREER; Geometric; Methods; Hyperbolic; Partial

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.

我们许多最著名的物理理论都是基于波浪般的部分微分方程（PDE）。重要的例子包括一般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。尽管数百年的数学进步，但我们对解决方案的理解仍然存在重大差距。例如，尽管预计解决方案经常“成为无限”，但数学家发现证据的情况很少。而且，在其他情况下，这些方程是否具有解决方案，甚至还不知道。因此，我们仍然对许多古典物理方程的物理预测没有明确的理解。主要障碍是，除了在不切实际的简化假设下，上述方程非常困难。但是，在解决其中一些困难方面取得了一些进步。例如，最近的结果提供了对爱因斯坦方程解决方案解决方案的大爆炸形成的详细描述，以及欧拉和相关方程的解决方案中的冲击。该项目的主要目标是将这些结果扩展到其他与物理相关的方程式和制度。这项工作将为波浪的长期行为以及各种物理理论分解的方式提供新的见解。该研究项目涉及对准双曲线PDE的严格研究，并具有三个主要的研究目标。首先是在各种准线性波般的物理意义方面中提供奇异性形成的详细描述，并强调尽可能避免对称性假设。示例包括各种爱因斯坦 - 摩擦方程，欧拉方程和弹性方程。这项研究将通过扩大方程式和已知会导致爆炸的初始条件类别来扩大我们对奇异性形成的理解。第二个研究目标是证明一类与物理相关的准四线双曲线问题具有自由边界的局部良好性，方程的双曲线会退化。第三个是开发新的工具，以理解在对精确特征知识的上下文中具有多个特征的准双曲PDE的解决方案的行为。一个最终目标是使用这些工具来证明，对于此类方程，在适当的假设对非线性和数据的假设下，发生冲击形成并且是稳定的。 该提案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级本科生进行调查，PI计划监督本科研究项目。也有适合博士生的组件。此外，PI与三个共同组织者一起组织，一对针对高级本科生的暑期学校和来自美国及以后的初学者。 PI还将开发课程材料，可以在线公众免费提供。