Mathematical Sciences: Shock Stability in Systems that Change Type

在这个项目中，首席研究员将继续她在可改变类型的偏微分方程组的适定性方面的重要工作。这样的系统有激波类型的解，作为波的传播机制，即使在系统不再是双曲的部分区域中也是如此。这些类型的问题出现在气体的跨音速流动和多相流体通过多孔介质的流动中。特别是，首席研究员将使用正则化和摄动方法研究从双曲型到椭圆型变化的方程组中激波的稳定性。流体(液体和气体)在不寻常条件下的行为，如在高层大气或地球中，由一个复杂的偏微分方程组控制，这种偏微分方程组可以发生所谓的“类型变化”。基本上，这意味着系统在域的不同部分有不同类型的解决方案。作为一个例子，想一想地震在地球内部产生的地震波的行为。波通过处于液体、固体或复合状态的区域。在这个项目中，首席研究人员将研究一个模型方程组的某些解，这些模型方程组表现出被称为“冲击波”的类型变化行为。冲击波的研究兴起于跨音速流动和多孔介质流动等重要技术领域。

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.