Mathematical Sciences: Continuum Mechanics and Hyperbolic Systems of Conservation Laws

研究人员将对连续体物理和双曲型守恒定律理论之间的接口问题进行研究。一个项目将涉及使用广义特征的方法来研究可能是也可能不是保守形式的拟线性双曲组的解的正则性、大时间行为和奇点的传播。另一个项目将包括对守恒律系统中波扇的可容许性的系统研究，对特定的物理现象进行模拟，其中可容许性的问题仍未解决，要么是由于相边界的存在，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通过“扩展热力学”产生的守恒定律系统，并研究将简单的理论作为特例或极限情况嵌入到更复杂的理论中的过程。连续介质物理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(例如，力学、热力学、电动力学)，而后者确定连续介质的类型(例如，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组合导致拟线性双曲组，通常被称为守恒律的双曲组。尽管在过去的25年里，这类系统的研究取得了长足的进步，但在一个空间维的情况下，建立可容许解的存在性、唯一性、正则性和大时间性态的基本目标仅部分地实现了，而在更高的空间维上仍然完全没有实现。这一领域的进步需要理解连续介质物理、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.