Viscosity and Relaxation Approximations of Hyperbolic Systems

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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.