Mathematical Sciences: Shock Stability in Systems that Change Type

在这个项目中，首席研究员将继续她的工作。 重要工作的适定性系统的部分 可以改变类型的微分方程。 这样的系统具有 冲击波型的解决方案，作为机制， 波的传播，即使在部分领域， 系统不再是双曲线的。 这些类型的问题发生在 气体的跨音速流动和多相流体的流动 通过多孔介质。 特别是，首席研究员 将研究方程组中冲击波的稳定性 从双曲线型变为椭圆型， 正则化和扰动方法。 流体（液体和气体）在异常条件下的行为 例如在高层大气或地球上的条件是 由一个复杂的偏微分系统控制 方程可以进行所谓的“类型改变”。 基本上，这意味着系统具有不同的解决方案， 在不同的领域的不同部分。 作为说明， 想象一下地震波的行为， 地球因地震。 波穿过的区域 液态、固态或复合态。 本工程 首席研究员将检查系统的某些解决方案 的模型方程，表现出改变类型的行为， 被称为“冲击波”。 冲击波的研究是在这样的 跨音速流动和流动等重要技术领域 通过多孔介质。

In this project the principal investigator will continue her important work on the well-posedness of systems of partial differential equations that can change type. Such systems have solutions of shock-wave type that serve as mechanisms for the propoagation of waves, even in parts of the domain where the system is no longer hyperbolic. These types of problems occur in the transonic flow of gases and in the flow of multi-phase fluids through porous media. In particular, the principal investigator will study the stability of shock waves in systems of equations that undergo changes from hyperbolic to elliptic type using regularization and perturbation methods. The behavior of fluids (liquids and gases) under unusual conditions such as in the upper atmosphere or in the earth is governed by a complicated system of partial differential equations that can under go what is called a "change of type". Basically this means that the system has solutions of different kinds in different parts of the domain. As an illustration, think of the behavior of a seismic wave generated inside the earth by a earthquake. The wave passes through regions that are in liquid, solid or composite states. In this project the principal investigator will examine certain solutions of a system of model equations that exhibit change-of-type behavior that are known as "shock waves". The study of shock waves arises in such technologically important areas as transonic flow and flow through porous media.