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

粘性守恒定律和完全可积系统中的点法和半群法

• 批准号: 0230003
• 负责人: Peter Howard
• 金额: $ 9.3万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-09-01 至 2004-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0230003&HistoricalAwards=false
• 关键词: Pointwise; Semigroup; Methods; Viscous; Conservation

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.

粘性守恒定律出现在各种物理应用中，包括流体动力学、磁流体动力学和材料科学。 特别重要的是此类方程的解是稳定的，因此通常与可观察的现象相对应。 不幸的是，事实证明，建立这些解决方案的稳定性是一个相当困难的问题。 然而，由刘发起并由刘和他的合作者开发的点格林函数方法已被证明相当稳健：在任意阶单一守恒定律中产生的粘性冲击波的应用中，在具有二阶扩散的系统中产生的粘性冲击波，平面粘性冲击波，简并粘性冲击波和稀疏波。 我们建议在三个方向上继续并扩展这一有前景的研究方向。 首先，Howard 和 Zumbrun 最近开发的新技术似乎适合扩展到（i）允许简并粘性冲击波的粘性守恒定律系统，以及（ii）具有高阶粘性的粘性守恒定律系统。 其次，我们建议开发进一步的技术，将点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微扰理论背景下开发的新技术纳入对二阶以上粘性守恒定律中出现的必然振荡动力学的研究中。能量和动量等基本属性的守恒通常会导致偏微分方程对某些潜在的物理过程进行建模。 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这个范式。 主要关注的是稳定现象：其主要结构对较小的环境波动具有鲁棒性。 我们建议继续并扩展一项有希望的研究路线，该研究路线在为这种稳定性建立明确的标准方面取得了巨大的成功。 该方法的直接结果是对某些基本偏微分方程的详细理解。