Stability of Patterns

摘要lin和Schecter提出用守恒律系统的Dafermos正则化来解决粘性守恒律系统的难题。前者是人为的数学构造；后者在科学中无处不在，在许多情况下，它们代表质量、动量、能量等的守恒。在他们早期工作的基础上，Lin和Schecter建议完成他们对线性化Dafermos算子谱的分析。他们建议使用这种分析来确定Riemann解作为粘性守恒律的渐近状态的稳定性。他们还建议研究在此工作过程中出现的相关问题，包括几何奇异摄动理论的交换引理的可能推广；守恒定律的三阶和四阶正则化的扩展以及无黏性守恒律系统黎曼解稳定性的新方法。在许多科学和技术领域，涉及流体流动的各种情况，如石油开采和制造中使用的薄液体膜的流动，可以用称为粘性守恒定律的方程进行数学建模。当一个人去掉各种术语，只留下一个守恒定律系统时，模型变得更容易处理。对于这些方程，人们通常可以构造显式解，称为黎曼解，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一侧主要是水，另一侧主要是油的移动锋；水把石油推向油井。黎曼解很重要的一个原因是，人们相信在许多情况下，黏性守恒定律的解，经过适当的缩放，随着时间的推移，会越来越像黎曼解。然而，这种行为只有在少数人为的情况下才能被证明。一个相关的事实是，我们没有很好的数学技术来检查黎曼解是否稳定，即，是否真的接近一组重要的粘性守恒定律的初始构型。Lin和Schecter开发了一种解决这些问题的新方法，使用了粘性守恒定律的不同简化，即所谓的Dafermos正则化。这个方程允许一个平滑版的黎曼解作为稳态。原则上，我们可以用比较熟悉的数学方法来检验它的稳定性。Lin和Schecter计划继续研究这些光滑黎曼解的稳定性，并利用这项工作来研究物理上的相关情况。

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.