Mathematical Sciences: Continuum Mechanics and Hyperbolic Systems of Conservation Laws

研究者将对连续介质物理和守恒定律双曲系统理论之间的界面问题进行研究。一个项目将涉及使用广义特征的方法来研究可能或可能不是保守形式的拟线性双曲系统解的正则性，大时间行为和奇点的传播。另一个项目将包括对守恒定律系统中的波扇可容许性的系统研究，模拟特定的物理现象，其中可容许性的问题仍然没有解决，要么是因为相边界的存在，要么是因为严格的双曲性失败。最后，第三个项目将涉及考虑守恒定律系统的层次结构，比如由玻尔兹曼方程或“扩展热力学”产生的系统，并研究简单理论作为特殊或极限情况嵌入更复杂理论的过程。连续介质物理学建立在平衡定律和本构关系的基础上。前者确定理论的框架（如力学、热力学、电动力学），后者确定连续介质的类型（如弹性、粘弹性、固体、流体）。这里的问题涉及具有“弹性”响应的介质，在这种情况下，平衡定律和本构关系的结合导致拟线性双曲系统，通常称为双曲守恒定律系统。尽管在过去的25年中，这类系统的研究取得了相当大的进展，但在一维空间中，建立可容许解的存在性、唯一性、正则性和大时间行为的基本目标仅部分实现，而在更高的空间维度中则完全没有实现。该领域的进步需要理解连续介质物理学、数学分析和科学计算的结合。

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.