Viscosity and Relaxation Approximations of Hyperbolic Systems

双曲系统的粘度和松弛近似

• 批准号: 9971934
• 负责人: Athanasios Tzavaras
• 金额: $ 8.9万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2002-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971934&HistoricalAwards=false
• 关键词: Viscosity; Relaxation; Approximations; Hyperbolic; Systems

This research is concerned with several aspects of the theory of weak solutions for hyperbolic systems, with the objective to developtechniques for understanding the emergence of shock waves from smooth approximate solutions in the small viscosity and small relaxation-time limits. Such questions are intimately tied to the mechanical issue of the passage from one thermomechanical theory to another, and appear in models of continuum physics, kinetic theory, and in the interface of the two. The goal is on the one hand to exploit ideas from the kinetic theory of gases in developing theory for hyperbolic systems, on the other hand in applying recent advances from the theory of hyperbolic equations in studying the passage from microscopic theories (of interacting particles) to mesoscopic theories (at the kinetic level) to macroscopic (continuum) theories. Shock waves are coherent structures that appear in situations involving high speed supersonic flows. Their mathematical modeling involves nonlinear hyperbolic partial differential equations. Understanding the theory of these equations is very important in the design and implementation of numerical algorithms for computing. The emphasis of this proposal is in situations involving shock waves in transitions from rarefied to dense gases. A typical application is air flow in high altitude supersonic flight,for example during reentry of a space shuttle into the atmosphere.

本研究关注双曲型方程组弱解理论的几个方面，目的是发展理解在小粘性和小松弛时间限制下从光滑近似解中出现激波的技术。 这些问题与从一个热力学理论过渡到另一个热力学理论的力学问题密切相关，并出现在连续介质物理学、动力学理论的模型中，以及两者的界面上。目标是一方面利用气体动力学理论的思想发展双曲系统的理论，另一方面应用双曲方程理论的最新进展研究从微观理论（相互作用粒子）到介观理论（在动力学水平）到宏观（连续）理论的通道。激波是相干结构，出现在涉及高速超音速流的情况下。其数学模型涉及非线性双曲型偏微分方程。理解这些方程的理论对于设计和实现数值计算算法是非常重要的。 这个建议的重点是在涉及从稀薄到稠密气体过渡的激波的情况下。 一个典型的应用是高空超音速飞行中的空气流动，例如在航天飞机重返大气层期间。