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

该项目将研究一系列波传播理论中的关键开放性问题，包括薄膜流动、模式形成、爆轰和气体动力学理论。所考虑的问题具有计算复杂性、多个长度/时间尺度以及应用程序中真正的物理兴趣等特征。许多问题是目前的数字和实验不足以解决的。计划中的几个子项目涉及使用保证误差范围的科学计算进行数值辅助证明，直至并包括严格的数值证明。该项目的一个组成部分是同时开发一个用户友好的数值平台，STABLAB，用于数值稳定性研究，并系统地探索在反应或电离流的微妙情况下气体和流体动力学的物理行为。所涉及的问题涉及动力系统、奇异摄动理论、线性算子的谱理论、非线性偏微分方程和严格的科学计算，并应导致开发新的通用数学工具。特别是，动力学激波和边界层问题的动力系统工具的发展将统一和扩展soone等人的“京都学派”使用各种形式和分析方法对玻尔兹曼现象获得的结果。同样，引入新的无粘稳定性标准的横摇波和Kreiss对称技术的调制前沿分析开辟了新的方向，在研究周期调制。关于疾驰爆炸的问题，如果得到解决，将回答一个长期存在的问题，而相关的严格的Wensel，Kramers和Brillouin方法的发展将具有广泛的普遍用途。浅水流中横摇波的简单稳定性判据的确定在水利工程中具有实际意义。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，对科学计算的标准具有更广泛的影响。

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.