Collaborative Research: Fundamental challenges in nonlinear hyperbolic PDE

合作研究：非线性双曲偏微分方程的基本挑战

• 批准号: 1311353
• 负责人: Helge Jenssen
• 金额: $ 36.06万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1311353&HistoricalAwards=false
• 关键词: Collaborative; Research; Fundamental; challenges; nonlinear

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.

尽管最近的进展，在非线性演化问题的分析，我们缺乏一个很好的数学理解波动的可压缩流体远离平衡。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，即使对于基本模型，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个合理的初始数据类的解的存在性理论。一个关键的问题是开发提供定量和定性预测的方法。通过寻求提供超越抽象存在结果的信息的方法，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双曲守恒律系统。作为对初值问题存在性结果的补充，我们还寻求此类系统的一般结构性质。其目的是理解一个系统的“几何”，如编码在其特征值，本征框架，熵等，影响溶液的性质。该项目的这一部分导致了几何学中的独立兴趣问题，并阐明了物理模型中底层结构的作用。考虑以下情况：一个完美的球形激波在收敛激波内静止的气体中向内传播。从实验和基本的考虑来看，很明显，当激波接近运动中心时，它通常会加速和加强。非常接近中心的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、气象学、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。这些模型通常被公式化为非线性方程组。这样的系统在自然现象的建模中无处不在，尤其是与流体流动有关-它们需要很好的理解。这种理解的数学方面提供了严格的结果有关的解决方案的存在性，其唯一性和稳定性，以及它们的定性性质。该提案解决了这些类型的基本问题的非线性现象，起源于可压缩气体流动模型。我们考虑的问题似乎是必须克服的基本路障，以获得正确的理解流体流动中的非线性现象。