Mathematical Sciences: Continuum Mechanics and Hyperbolic Systems of Conservation Laws

调查员将对存在的问题进行研究 在连续介质物理学和 双曲守恒律组 一个项目将 涉及采用广义特征线方法， 研究了地震波传播规律、大时间行为及传播规律 拟线性双曲方程组解的奇性， 可以是也可以不是保守的形式。 另一个项目将 包括对波扇可容性的系统研究 在守恒定律系统中， 在这种情况下，可否受理问题仍然是一个问题。 由于相界的存在或 因为严格双曲性失败了。 最后，第三个项目 将涉及到考虑保护系统的层次 定律，如玻尔兹曼方程或通过 “扩展热力学”，并研究 更简单的理论作为特殊或限制情况嵌入到 更复杂的。 连续统物理学是建立在平衡定律和 本构关系 前者决定了 理论（例如，力学，热力学， 电动力学），而后者确定的类型， 连续介质（例如弹性的，粘弹性的，固体的， 流体）。 这里的问题涉及媒体的“弹性”反应 在这种情况下，平衡定律和本构定律的结合 关系导致准线性双曲系统，通常称为 作为双曲守恒律系统。 尽管有相当大 过去25年来，这类系统的研究取得了进展， 基本目标是建立存在性、唯一性、规律性和 容许解的大时间行为仅被 在一个空间维度的情况下部分实现， 在更高的空间维度中完全无法实现。 进展 这一领域需要理解连续介质物理学的结合， 数学分析和科学计算。

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.