Viscosity and Relaxation Approximations of Hyperbolic Systems

本研究涉及双曲系统弱解理论的几个方面，目的是发展理解小粘度和小松弛时间极限下光滑近似解产生激波的技术。这些问题与从一种热力学理论过渡到另一种热力学理论的力学问题密切相关，并出现在连续介质物理模型、动力学理论和两者的界面中。我们的目标一方面是利用气体动力学理论的思想来发展双曲系统的理论，另一方面是应用双曲方程理论的最新进展来研究从微观理论（相互作用的粒子）到介观理论（在动力学水平上）到宏观（连续体）理论的过渡。激波是在涉及高速超音速流动的情况下出现的相干结构。他们的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在计算数值算法的设计和实现中是非常重要的。这一建议的重点是在激波从稀薄气体过渡到稠密气体的情况下。一个典型的应用是高空超音速飞行中的气流，例如在航天飞机重返大气层时。

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.