Mathematical Sciences: Continuum Mechanics and Hyperbolic Systems of Conservation Laws

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The investigator will conduct research on the problems lying on the interface between continuum physics and the theory of hyperbolic systems of conservation laws. One project will involve employing the method of generalized characteristics to study regularity, large time behavior and propagation of singularities in solutions of quasilinear hyperbolic systems that may or may not be in conservative form. Another project will consist of a systematic study of the admissibility of wave fans in systems of conservation laws, modeling specific physical phenomena, in which the issue of admissibility is still unresolved either due to the presence of phase boundaries or because strict hyperbolicity fails. Finally, a third project will involve considering hierarchies of systems of conservation laws, like those generated by the Boltzmann equation or through "extended thermodynamics," and studying the process by which simpler theories are embedded as special or limiting cases into more complex ones. Continuum physics is founded on balance laws and constitutive relations. The former determine the framework of the theory (for example, mechanics, thermodynamics, electrodynamics) while the latter identify the type of the continuous medium (for example, elastic, viscoelastic, solid, fluid). The problems here concern media with "elastic" response in which case the combination of balance laws and constitutive relations leads to quasilinear hyperbolic systems, commonly known as hyperbolic systems of conservation laws. Despite considerable progress in the study of such systems over the past 25 years, the basic goal of establishing existence, uniqueness, regularity and large time behavior of admissible solutions has only been achieved partially in the case of one space dimension and remains entirely unfulfilled in higher space dimensions. Progress in the field requires understanding a combination of continuum physics, mathematical analysis, and scientific computation.