New Tools in the Study of Wave Propagation: Dynamical Systems for Kinetic Equations, Inviscid Limits for Modulated Periodic Waves, and Rigorous Numerical Stability Analysis

波传播研究的新工具：运动方程的动力系统、调制周期波的无粘极限以及严格的数值稳定性分析

• 批准号: 1700279
• 负责人: Kevin Zumbrun
• 金额: $ 20.8万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700279&HistoricalAwards=false
• 关键词: New; Tools; Study; Wave; Propagation

The PI will study a selection of key open problems in the theory of wave propagation arising in a variety of settings including thin film flow, pattern formation, detonation, and the kinetic theory of gases. The problems considered share the features of computational complexity, multiple length/time scales, and genuine physical interest in applications. Many concern questions that current numerics and experiment are not adequate to resolve. Several of the planned subprojects involve numerically assisted proof using scientific computation with guaranteed error bounds, up to and including rigorous numerical proof. An integral part of the project is the simultaneous development of a user-friendly numerical platform, STABLAB, for numerical stability investigation, and the systematic exploration with this platform of physical behavior in gas and fluid dynamics in the delicate situations of reacting or ionized flow. The problems addressed involve issues in dynamical systems, singular perturbation theory, spectral theory of linear operators, nonlinear partial differential equations, and rigorous scientific computation, and should result in the development of new mathematical tools of general application. In particular, development of dynamical systems tools for kinetic shock and boundary layer problems would unify and extend results obtained for Boltzmann phenomena by the "Kyoto School" of Sone et al using a variety of formal and analytic methods. Likewise, the introduction of new inviscid stability criteria for roll waves and of Kreiss symmetrizer techniques for analysis of modulated fronts open new directions in the study of periodic modulation. The problem on galloping detonations, if solved, will answer a longstanding question, while associated rigorous Wensel,Kramers, and Brillouin method developments will be of wide general use. Determination of simple stability criteria for roll waves in shallow water flow are of practical interest in hydraulic engineering. Finally, the development of rigorous numerical proof and error estimate techniques is potentially transformative, having broader implications for standards in scientific computing.

PI将研究在各种环境中引起的波传播理论中的关键开放问题，包括薄膜流，模式形成，爆炸和气体动力学理论。 这些问题共享计算复杂性，多长度/时间尺度以及对应用程序的真正身体兴趣的特征。 许多关注当前数字和实验不足以解决的问题。 一些计划的子项目涉及使用科学计算和保证的误差界限的数字辅助证明，直到和包括严格的数值证明。 该项目不可或缺的一部分是，在反应或电离流的微妙情况下，同时开发了用户友好的数值平台，用于数值稳定性调查，以及与气体和流体动力学平台的系统探索。 解决的问题涉及动态系统中的问题，奇异扰动理论，线性操作员的光谱理论，非线性偏微分方程以及严格的科学计算，并应导致开发新的一般应用数学工具。特别是，使用多种形式和分析方法的“京都学校”的“京都学校”的“京都学校”统一和扩展了用于统一的动力学系统工具的动力学系统工具。同样，在定期调制的研究中，引入了滚动波和Kreiss对称器技术的新的无粘性稳定标准和Kreiss对称器技术进行分析。疾驰的爆炸问题（如果解决）将回答一个长期存在的问题，而相关的严格Wensel，Kramers和Brillouin方法的开发将广泛使用。确定浅水流中滚动波的简单稳定性标准在液压工程中具有实际感兴趣。最后，严格的数值证明和误差估计技术的发展具有潜在的变革性，对科学计算中的标准具有更广泛的影响。

项目成果

A calculus proof of the Cramér–Wold theorem

CraméräWold 定理的微积分证明

• DOI: 10.1090/proc/13794
• 发表时间: 2018
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Lyons, Russell;Zumbrun, Kevin
• 通讯作者: Zumbrun, Kevin

Turing patterns in parabolic systems of conservation laws and numerically observed stability of periodic waves

守恒定律抛物线系统中的图灵模式和周期波的数值观测稳定性

• DOI: 10.1016/j.physd.2017.12.003
• 发表时间: 2018
• 期刊: Physica D: Nonlinear Phenomena
• 作者: Barker, Blake;Jung, Soyeun;Zumbrun, Kevin
• 通讯作者: Zumbrun, Kevin

Multidimensional Stability of Large-Amplitude Navier–Stokes Shocks

• DOI: 10.1007/s00205-017-1147-7
• 发表时间: 2017-07
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: J. Humpherys;Gregory Lyng;K. Zumbrun

Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks

一类奇异演化方程的稳定流形和运动激波的指数衰减

• DOI: 10.3934/krm.2019001
• 发表时间: 2019
• 期刊: Kinetic & Related Models
• 影响因子: 1
• 作者: Pogan, Alin;Zumbrun, Kevin
• 通讯作者: Zumbrun, Kevin

Stability of Viscous St. Venant Roll Waves: From Onset to Infinite Froude Number Limit

粘性圣维南滚波的稳定性：从起始到无限弗劳德数极限

• DOI: 10.1007/s00332-016-9333-6
• 发表时间: 2017
• 期刊: Journal of Nonlinear Science
• 影响因子: 3
• 作者: Barker, Blake;Johnson, Mathew A.;Noble, Pascal;Rodrigues, L. Miguel;Zumbrun, Kevin
• 通讯作者: Zumbrun, Kevin