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良好的数学理解。这是一个在许多应用中具有明显相关性的制度。令人沮丧的是，即使是基本模型，人们对它们的有效性范围也知之甚少。数学方法的第一步是寻求一种合理的初始数据类的解的存在性理论。一个关键问题是开发既能提供定量预测又能提供定性预测的方法。通过寻找给出抽象存在结果以外的信息的方法，我们寻求评估常规用于实际应用的模型的相关性和局限性，例如可压缩欧拉流。我们所考虑的模型被表述为守恒律的双曲型系统。作为初值问题存在性结果的补充，我们还研究了这类系统的一般结构性质。其目的是理解系统的“几何”如何影响解的性质，如编码在其特征值、特征框架、熵等中。该项目的这一部分引出了对几何学的独立兴趣问题，并阐明了物理模型中底层结构的作用。考虑以下场景：一个完美的球形激波在静止在会聚激波内的气体中向内传播。从实验和基本考虑来看，很明显，当激波接近运动中心时，它通常会加速并增强。在离中心非常近的地方，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、燃烧等应用中都有很大的兴趣。然而，我们目前对这种类型的流体流动缺乏很好的了解，原因是缺乏数学洞察力。描述物理过程的基本模型已经有150多年的历史了。尽管如此，我们仍在寻找甚至是根本问题的答案。这些模型通常被表述为非线性方程组。这样的系统在自然现象的建模中无处不在，最重要的是与流体流动有关-它们需要很好的理解。这种理解的数学方面是关于解的存在、解的唯一性和稳定性以及解的定性性质的严格结果。该提案解决了源于可压缩气体流动模型的这些类型的非线性现象的基本问题。我们考虑的问题似乎是必须克服的基本障碍，以便正确理解流体流动中的非线性现象。

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.