Hyperbolic and Kinetic Partial Differential Equations

NSF奖摘要-DMS-0205032数学科学：双曲和动力学偏微分方程式这个项目涉及双曲型系统弱解理论的几个方面，以及气体运动论中出现的输运方程的数学理论。具体主题是：(I)探索双曲型系统弱解理论和气体运动论中输运方程理论之间的接口，特别是关于振荡的传播和消除的问题。(2)研究某些碰撞动力学模型的适定性和流体动力学极限，这些问题与微观理论向连续介质理论过渡过程中产生的热力学问题密切相关。(3)利用变分技术研究多维弹性动力学和粘弹性方程的结构性质。(4)分析扩散敏感动力学的各种情况，如小粘度对双曲系统长期演化的影响以及扩散敏感系统的图解概念。双曲型守恒定律的数学研究在很大程度上是由物理学和连续介质力学中的基本守恒定律推动的。从该领域的早期阶段开始，分析方法和数值方法就一起发展起来，而分析理解有助于设计高性能的计算算法。近年来，动力学方程理论和双曲组弱解理论之间的思想交流卓有成效。这种交流的核心是从气体运动理论或统计物理的微观模型中推导出连续统理论的问题。这个项目将利用这种思想交流来开发数学技术，以更好地理解由守恒定律建模的各种重要物理系统。

NSF Award Abstract - DMS-0205032Mathematical Sciences: Hyperbolic and kinetic partial differential equationsAbstract0205032 TzavarasThis project deals with several aspects of the theory of weak solutions for hyperbolic systems, and the mathematical theory of transport equations that arise in kinetic theory of gases.Specific themes are:(i) To exploit the interface between the theory of weak solutions for hyperbolic systems and the theory of transport equations in the kinetic theory of gases, particularly with regard to issues of propagation and cancellation of oscillations. (ii) To study well-posedness and hydrodynamic limits for certain collisional kinetic models, problems that are intimately connected to the thermomechanical issues arising in the passage from microscopic to continuum theories. (iii) To exploit variational techniques in the study of the structural properties for the equations of multi-dimensional elastodynamics and viscoelasticity. (iv) To analyze various instances of diffusion-sensitive dynamics, like the effect of small-viscosity on the long-time evolution of hyperbolic systems and the notion of graph solutions for diffusion sensitive systems.The mathematical research on hyperbolic systems of conservation laws is to a large extent motivated by the fundamental conservation laws in physics and continuum mechanics. From its early stages, analytical and numerical methods in this field have developed together, and analytical understanding contributes in the design of high performance computational algorithms. In recent years there has seen a very fruitful exchange between ideas in the theory of kinetic equations and the theory of weak solutions for hyperbolic systems. At the core of this exchange is the issue of deriving continuum theories from microscopic models of kinetic theory of gases or statistical physics. This project will make use of this exchange of ideas to develop mathematical techniques to better understand the wide variety of important physical systems that are modeled by conservation laws.