Hyperbolic and Kinetic Partial Differential Equations

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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.