Pointwise and Semigroup Methods in Viscous Conservation Laws and Completely Integrable Systems

粘性守恒定律出现在各种各样的物理应用中，包括流体力学、磁流体力学和材料科学。特别重要的是这些方程的解是稳定的，因此通常与可观察到的现象相对应。不幸的是，建立这些解决方案的稳定性已被证明是一个相当困难的问题。然而，由刘和他的合作者发起并发展的点向格林函数方法已经被证明是相当鲁棒的：在应用于任意阶的单一守恒定律中产生的粘性激波，二阶扩散系统中产生的粘性激波，平面粘性激波，简并粘性激波和稀薄波。我们建议在三个方向上继续和扩展这一有前途的研究方向。首先，Howard和Zumbrun最近开发的新技术似乎适合扩展到(i)允许退化粘性激波的粘性守恒律系统，以及（ii）具有高阶粘性的粘性守恒律系统。其次，我们建议开发进一步的技术，将点向格林函数方法扩展到粘性稀疏波的情况。最后，我们希望将最近在完全可积系统的摄动理论背景下发展起来的新技术纳入到研究在高于二阶的粘性守恒定律中产生的必然振荡动力学中。像能量和动量这样的基本性质的守恒常常导致偏微分方程来模拟一些潜在的物理过程。例如，流体动力学的纳维-斯托克斯方程和电磁学的麦克斯韦方程都遵循这种范式。主要关注的是稳定现象：那些其主要结构对轻微的环境波动是稳健的。我们建议继续并扩大一项有前途的研究，这项研究在确立这种稳定性的明确标准方面非常成功。这种方法的直接结果是对某些基本偏微分方程的详细理解。

Viscous conservation laws arise in a wide variety of physical applications, including fluid dynamics, magnetohydrodynamics,and materials science. Of particular importance are solutionsof such equations that are stable and hence typically correspond with observable phenomena. Unfortunately, establishingthe stability of these solutions has proven to be a quitedifficult problem. The pointwise Green's function approach,however, initiated by Liu and developed by Liu and his collaborators, has proven quite robust: in applications to viscous shock waves arising in single conservation laws ofarbitrary order, viscous shock waves arising in systems with second order diffusion, planar viscous shock waves, degenerate viscous shock waves, and rarefaction waves. We propose to continue and extend this promising line ofresearch in three directions. First, new techniques recently developed by Howard and Zumbrun appear suitablefor extension to (i) systems of viscous conservation laws admitting degenerate viscous shock waves, and (ii)systems of viscous conservation laws with high orderviscosity. Second, we propose to develop further techniques that will extend the pointwise Green's function approach to the case of viscous rarefaction waves. Finally, we would like to incorporate new techniquesrecently developed in the context of perturbation theoryfor completely integrable systems into the study of the necessarily oscillatory dynamics that arise in viscous conservation laws of order higher than two.The conservation of such fundamental properties as energy andmomentum often leads to partial differential equationsthat model some underlying physical process. For example,the Navier-Stokes equations of fluid dynamics and the Maxwell equations of electromagnetism follow this paradigm. Of primary concern are stable phenomena: thosewhose principal structure is robust to minor environmentalfluctuations. We propose to continue and extend a promising line of research that has been extraordinarily successful in establishing a clear criterion for suchstability. A direct consequence of the approach is a detailed understanding of certain fundamental partialdifferential equations.