Stability of Patterns

摘要林和谢克特提出利用守恒律系统的Dafermos正则化来解决粘性守恒律系统的困难问题。前者是一个人工的数学概念；后者在科学中普遍存在，在许多情况下它们代表质量、动量、能量等守恒。在他们早期工作的基础上，Lin和Scheck建议完成对线性化Dafermos算子的频谱分析。他们建议利用这种分析来确定作为粘性守恒律的渐近态的Riemann解的稳定性。他们还建议研究在这项工作中出现的相关问题，包括几何奇异摄动理论的交换引理的可能推广；守恒律的三阶和四阶正则化的推广；以及无粘性守恒律组Riemann解的稳定性的新方法。在许多科学和技术领域，涉及流体流动的各种情况，如石油回收和制造中使用的薄膜液体流动，可以用称为粘性守恒律的方程来数学建模。当人们去掉不同的项时，模型变得更容易处理，只剩下一个守恒定律体系。对于这些方程，人们通常可以构造出称为黎曼解的显式解，它经常涉及以不同速度移动的跳跃。注水采油的一个例子是移动的前锋，一侧主要是水，另一侧主要是石油；水将石油推向油井。黎曼解很重要的一个原因是，人们认为，在许多情况下，粘性守恒律的解，经过适当的重新标度，随着时间的推移，往往看起来越来越像黎曼解。然而，只有少数几种相当人为的情况可以证明这一行为。一个相关的事实是，我们没有很好的数学技术来检查Riemann解是否稳定，即是否真正接近于粘性守恒律的一组重要的初始构型。林和谢克特开发了一种新的方法来解决这些问题，他们使用了对粘性守恒定律的不同简化，即所谓的Dafermos正则化。这个方程允许黎曼解的一个平滑版本作为稳态。原则上，人们可以用相对熟悉的数学方法来检验它的稳定性。林和谢克特计划继续他们关于这些光滑黎曼解的稳定性的工作，并利用这项工作来接近物理上相关的情况。

AbstractLin and Schecter propose to use the Dafermos regularization of a system of conservation laws to approach difficult questions concerning systems of viscous conservation laws. The former is an artificial mathematical construct; the latter are ubiquitous in the sciences, where they represent conservation of mass, momentum, energy, etc. in many situations. Building on their earlier work, Lin and Schecter propose to complete their analysis of the spectrum of the linearized Dafermos operator. They propose to use this analysis to determine the stability of Riemann solutions as asymptotic states of viscous conservation laws. They also propose to investigate related issues that have arisen in the course of this work, including possible generalizations of the Exchange Lemma of geometric singular perturbation theory; extensions to third- and fourth-order regularizations of conservation laws; and a new approach to stability of Riemann solutions of systems of conservation laws without viscosity.In many areas of science and technology, various situations involving fluid flow, such as oil recovery and flow of thin liquid films used in manufacturing, can be mathematically modeled by equations called viscous conservation laws. The models become more tractable when one drops various terms, leaving only a system of conservation laws. For these equations one can often construct explicit solutions called Riemann solutions, that frequently involve jumps that move with varying speeds. An example from oil recovery using injection of water is a moving front that is mostly water on one side and mostly oil on the other; the water pushes the oil toward the well. One reason Riemann solutions are important is that it is believed that in many situations, solutions of viscous conservation laws, appropriately rescaled, tend to look more and more like Riemann solutions as time goes on. However, there are only a few, rather artificial situations is which this behavior is proved. A related fact is that we do not have good mathematical techniques to check whether Riemann solutions are stable, i.e., are really approached for a significant set of initial configurations of the viscous conservation laws. Lin and Schecter have developed a new approach to these issues using a different simplification of the viscous conservation laws, the so-called Dafermos regularization. This equation admits a smoothed-out version of the Riemann solution as a steady-state. In principle, one can check its stability by relatively familiar mathematical methods. Lin and Schecter plan to continue their work on the stability of these smoothed Riemann solutions, and to use this work to approach the physically relevant situation.