Collaborative Research: Fundamental challenges in nonlinear hyperbolic PDE

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Despite recent progress in the analysis of non-linear evolutionary problems we lack a good mathematical understanding of wave motion in compressible fluids far from equilibrium. This is a regime of obvious relevance in many applications. Even for basic models frustratingly little is known rigorously about their range of validity. A first step in a mathematical approach is to pursue a theory of existence of solutions for a reasonable class of initial data. A key concern is to develop methods that provide quantitative and qualitative predictions as well. By pursing methods that give information beyond abstract existence results we seek to assess the relevance and limits of models that are routinely used for practical applications, such as compressible Euler flow. The models we consider are formulated as hyperbolic systems of conservation laws. As a complement to existence results for initial value problems we also seek general structural properties of such systems. The aim is an understanding of how the "geometry" of a system, as encoded in its characteristic values, eigen-frame, entropies, etc., impact the properties of solutions. This part of the project leads to questions of independent interest in geometry, and also clarifies the role of underlying structures in physical models. Consider the following scenario: a perfectly spherical shock wave propagates inward in a gas which is at rest within the converging shock. From experiments and basic considerations it is clear that the shock will typically accelerate and strengthen as it approaches the center of motion. Very close to the center the gas will experience enormous densities, velocities and temperatures. This and similar physical situations are of great interest in applications such as high-speed flight, meteorology, combustion, etc. However, we currently lack a good understanding of this type of fluid flow, and the reason for this is a lack of mathematical insight. The basic models for describing the physical processes have been known for more than 150 years. Nonetheless, we are still searching for answers to even fundamental questions. These models are typically formulated as systems of non-linear equations. Such systems are ubiquitous in modeling of natural phenomena, and above all in connection with fluid flow - they demand a good understanding. The mathematical aspect of such understanding is provided by rigorous results pertaining to existence of solutions, their uniqueness and stability, and their qualitative properties. The proposal addresses these types of fundamental issues for non-linear phenomena that originate in models for compressible gas flow. The problems we consider appear to be essential road blocks that must be overcome in order to gain a proper understanding of non-linear phenomena in fluid flow.