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

粘性守恒定律在许多物理应用中都有应用，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，由刘和他的合作者提出并发展的逐点格林函数方法已经被证明是相当稳健的：在应用于由任意阶单守恒律引起的粘性激波时，粘性激波出现在具有二阶扩散的系统中，平面粘性激波，简并粘性激波和稀疏波。我们建议在三个方向上继续和扩展这条充满希望的研究路线。首先，Howard和Zumbrun最近发展的新技术似乎适合于推广到(I)允许简并粘性激波的粘性守恒律组，和(Ii)具有高阶粘性的粘性守恒律组。其次，我们建议发展进一步的技术，将逐点格林函数方法推广到粘性稀疏波的情况。最后，我们想把最近在完全可积系统的微扰理论背景下发展起来的新技术结合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。能量和动量等基本性质的守恒通常会导致偏微分方程组，它模拟了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就遵循了这一范例。主要关注的是稳定现象：其主要结构对微小的环境波动是健壮的。我们建议继续和扩展一系列有希望的研究，这些研究已经非常成功地建立了这种稳定性的明确标准。这种方法的一个直接结果是对某些基本偏微分方程式有了详细的了解。

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.