The Dafermos Regularization of a System of Conservation Laws

林和Schecter建议使用的Dafermos正规化系统的守恒律来解决困难的问题有关系统的粘性守恒律。 前者是一种人为的物理概念;后者在科学中无处不在，在许多情况下，它们代表了质量、动量、能量等守恒。 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作为粘性守恒律渐近状态的黎曼解的稳定性。 他们还建议使用它来分析和改进基于Dafermos正则化的数值方法，用于计算黎曼解的曲线。 他们建议调查issuesin几何奇异摄动理论提出的Dafermosregularization。此外，Schecter和Lin将继续与合作者一起寻找带有反应项的粘性守恒定律的行波解;正在调查的主题包括液-汽相变和使用热量的采油方法。 在许多科学和技术领域，涉及流体流动的各种情况，如石油开采和制造业中使用的薄液膜流动，可以通过称为粘性守恒定律的方程进行数学建模。 当我们去掉各种项，只留下一个守恒定律系统时，模型就变得更容易处理了。 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。 使用注水采油的一个例子是一个移动的前沿，一边主要是水，另一边主要是油;水把油推向油井。 黎曼解很重要的一个原因是，人们相信在许多情况下，粘性守恒律的解，经过适当的重新标度，随着时间的推移，看起来越来越像黎曼解。然而，只有少数几个，而不是人为的情况是证明这种行为。 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，是真正接近的一组重要的初始配置的粘性守恒定律。 Lin和Schecter开发了一种新的方法来解决这些问题，使用粘性守恒律的不同简化，即所谓的Dafermos正则化。 这个方程允许黎曼解的平滑版本作为稳态。 原则上，人们可以通过相对熟悉的数学方法来检查其稳定性。 Lin和Schecter计划继续研究这些光滑黎曼解的稳定性，并利用这项工作来接近物理相关的情况。

Lin 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 mathematicalconstruct; 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 use it to analyze and improve Dafermos regularization-based numerical methods for computing curves of Riemann solutions. They propose to investigate issuesin geometric singular perturbation theory raised by the Dafermosregularization. In addition, Schecter and Lin will continue work withcollaborators on finding traveling wave solutions of viscous conservation laws with reaction terms; subjects under investigation include liquid-vapor phase transitions and methods of oil recovery that use heat. They will investigate the use of the Dafermos regularization in such problems.In many areas of science and technology, various situations involving fluid flow, such as oil recovery and flow of thin liquid films used inmanufacturing, 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 thewell. 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 configurationsof 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.