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.

粘性保护定律出现在各种物理应用中，包括流体动力学，磁性流失动力学和材料科学。 这种方程的解决方案尤其重要，因此通常与可观察到的现象相对应。 不幸的是，确立这些解决方案的稳定性已被证明是一个戒烟问题。 然而，格林的功能方法是由刘和他的合作者开发的，事实证明是很健壮的：适用于在降序的单一保护法中引起的粘性冲击波，在具有二阶粘性冲击波的系统中引起的粘性冲击波，粘性的冲击波，粘性冲击波，变态的冲击波，退化的粘性冲击波和稀有的浪潮和稀有浪潮。 我们建议继续并扩展这一有希望的途径，以三个方向发展。 首先，Howard和Zumbrun最近开发的新技术似乎适合扩展到（i）粘性保护法的系统，该系统承认粘性粘性冲击波，以及（ii）具有较高订单的粘性保护定律的系统。 其次，我们建议开发进一步的技术，以将绿色的官方方法扩展到粘性稀疏波的情况下。 最后，我们想结合在扰动理论的背景下开发的新技术，将完全可集成的系统用于研究一定要振荡的动力学，这些振荡动力学在粘性保护法中产生的稳定量高于两个高于两者的稳定定律。对诸如能量和摩姆的基本属性的保护通常会导致部分差分方程的物理过程，从而导致某些基础的物理过程。 例如，流体动力学的navier-stokes方程和电磁的麦克斯韦方程遵循此范式。 主要关注的是稳定的现象：这些主要的主要结构对较小的环境反弹是鲁棒的。 我们建议继续并扩展一项有希望的研究线，在建立这种稳定的明确标准方面取得了非常成功的成功。 该方法的直接结果是对某些基本偏差方程的详细理解。