Collaborative Research: Fundamental challenges in nonlinear hyperbolic PDEs

尽管最近在非线性演化问题的分析方面取得了进展，但我们对远离平衡的可压缩流体中的波动缺乏很好的数学理解。这是一个在许多应用程序中明显相关的制度。令人沮丧的是，即使是最基本的模型，人们对它们的有效范围也知之甚少。数学方法的第一步是为一类合理的初始数据寻求解的存在性理论。一个关键的问题是发展提供定量和定性预测的方法。通过追求超越抽象存在结果的信息的方法，我们试图评估实际应用中常规使用的模型的相关性和局限性，例如可压缩欧拉流。我们考虑的模型被表述为守恒定律的双曲系统。作为初值问题存在性结果的补充，我们还寻求这类系统的一般结构性质。其目的是理解系统的“几何”是如何以其特征值、特征框架、熵等编码影响解的性质的。项目的这一部分引发了对几何的独立兴趣问题，也阐明了物理模型中底层结构的作用。考虑下面的场景：一个完美的球形激波在气体中向内传播，而气体在会聚激波中处于静止状态。从实验和基本考虑可以清楚地看出，冲击在接近运动中心时通常会加速和加强。在非常靠近中心的地方，气体将经历巨大的密度、速度和温度。这种和类似的物理情况在高速飞行、气象、燃烧等应用中具有很大的兴趣。然而，我们目前对这种类型的流体流动缺乏很好的理解，其原因是缺乏数学洞察力。描述物理过程的基本模型在150多年前就已为人所知。尽管如此，我们仍在寻找一些基本问题的答案。这些模型通常被表述为非线性方程组。这种系统在自然现象的建模中无处不在，尤其是在与流体流动有关的建模中——它们需要很好的理解。这种理解的数学方面是由有关解的存在性、唯一性和稳定性以及定性性质的严格结果提供的。该提案解决了可压缩气体流动模型中产生的非线性现象的这些基本问题。为了正确理解流体流动中的非线性现象，我们所考虑的问题似乎是必须克服的重要障碍。

Despite recent progress in the analysis of non-linear evolutionary problems we lack a good mathematical understanding of wave motion in compressible fluids far from equilibrium. This is a regime of obvious relevance in many applications. Even for basic models frustratingly little is known rigorously about their range of validity. A first step in a mathematical approach is to pursue a theory of existence of solutions for a reasonable class of initial data. A key concern is to develop methods that provide quantitative and qualitative predictions as well. By pursing methods that give information beyond abstract existence results we seek to assess the relevance and limits of models that are routinely used for practical applications, such as compressible Euler flow. The models we consider are formulated as hyperbolic systems of conservation laws. As a complement to existence results for initial value problems we also seek general structural properties of such systems. The aim is an understanding of how the "geometry" of a system, as encoded in its characteristic values, eigen-frame, entropies, etc., impact the properties of solutions. This part of the project leads to questions of independent interest in geometry, and also clarifies the role of underlying structures in physical models. Consider the following scenario: a perfectly spherical shock wave propagates inward in a gas which is at rest within the converging shock. From experiments and basic considerations it is clear that the shock will typically accelerate and strengthen as it approaches the center of motion. Very close to the center the gas will experience enormous densities, velocities and temperatures. This and similar physical situations are of great interest in applications such as high-speed flight, meteorology, combustion, etc. However, we currently lack a good understanding of this type of fluid flow, and the reason for this is a lack of mathematical insight. The basic models for describing the physical processes have been known for more than 150 years. Nonetheless, we are still searching for answers to even fundamental questions. These models are typically formulated as systems of non-linear equations. Such systems are ubiquitous in modeling of natural phenomena, and above all in connection with fluid flow - they demand a good understanding. The mathematical aspect of such understanding is provided by rigorous results pertaining to existence of solutions, their uniqueness and stability, and their qualitative properties. The proposal addresses these types of fundamental issues for non-linear phenomena that originate in models for compressible gas flow. The problems we consider appear to be essential road blocks that must be overcome in order to gain a proper understanding of non-linear phenomena in fluid flow.