New problems in continuum mechanics: asymptotic eigenvalue distributions, rigorous numerical stability analysis and weakly nonlinear asymptotics in periodic thin film flow

Kevin Zumbrun建议解决一系列关键的公开问题，包括激波和爆震波的稳定性和行为，以及薄膜流动、光学和各种其他环境中出现的周期性图案。这些问题具有计算复杂性和在多个长度和时间尺度上发生的过程之间的微妙相互作用的特征，以及它们涉及基本的和频繁发生的物理现象的事实，几十年来人们已经进行了大量的研究，但在严格的数学水平上仍然没有解决。值得注意的是，这是一个例子，其中数学不仅仅是验证逻辑上已经观察到的物理或实验原理，而是在当前的数值和实验不足以解决的环境中找到秩序。计划中的几个子项目涉及使用科学计算进行数字辅助证明，并保证误差范围。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，并利用该平台系统地探索在反应或电离流动的微妙情况下气体和流体动力学中的物理行为。大多数已提出的计算，特别是那些同时涉及多维和粘性效应的计算，以前从未成功地进行过，因此该程序也有大量的数值/计算部分。所涉及的问题涉及转折点理论、非自伴算子的谱理论、非线性偏微分方程组和模式形成等有趣的和非标准的问题。所考虑的问题是长期存在的具有基本物理意义的问题，其解决方案将需要显著新的工具。特别是，严格的数值稳定性验证算法的发展；无界区域上解析系数转折点问题的处理，特别是无穷远处转折点的处理；粘性与高活化能组合的惊人效应的研究，似乎在流体和气体动力流动的稳定性和分叉的研究中可能是变革性的。这些问题中的每一个都涉及多重尺度(刚性)和没有光谱间隙的技术困难；它们的成功分析涉及通过固定相和相关的复分析方法在线性水平上的微妙抵消，以及在非线性水平上通过PI和合作者开发的相位提取/调制技术。严格的解析WKB理论、粘性爆轰理论和严格的数值稳定性验证(证明)的目标尤其具有变革性的潜力。同时，为没有数据的模型制作定量数据(例如，对爆轰稳定性的粘性影响)应立即具有实际用途。

Kevin Zumbrun proposes to attack a selection of key open problems in stability and behavior of shock and detonation waves and of periodic patterns arising in thin film flow, optics, and a variety of other contexts. These problems share the features of computational complexity and delicate interactions between processes occurring at multiple length and time scales, along with the fact that they concern fundamental and frequently occurring physical phenomena, have been much studied over a period of several decades, and yet at a rigorous mathematical level remain unresolved. It is important to note that this is an instance where mathematics is not just verifying logically already-observed physically or experimentally principles, but finding order in settings 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. 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. Most of the proposed computations, particularly those involving multiple dimensions and viscous effects simultaneously, have never before been successfully carried out- hence there is a substantial numerical/computational component to this program as well.The problems addressed involve interesting and nonstandard issues in turning point theory, spectral theory of nonselfadjoint operators, nonlinear partial differential equations, and pattern formation. The problems considered are long-standing ones of basic physical interest, whose solutions will require significantly new tools. In particular, development of rigorous numerical stability verification algorithms; treatment of analytic-coefficient turning point problems on unbounded domains, and especially with turning points at infinity; and the investigation of startling effects of viscosity in combination with high activation energy appear likely to be transformational in the study of stability and bifurcation of fluid- and gas-dynamical flow. Each of these problems involve the technical difficulties of multiples scales (stiffness) and absence of spectral gap; their successful analysis involves accounting of delicate cancellation both at the linear level, through stationary phase and related complex analytic methods, and at the nonlinear level, through phase extraction/modulation techniques developed by the PI and collaborators. The goals of rigorous analytic WKB theory, viscous detonation theory, and rigorous numerical stability verification (proof) in particular have the potential to be transformative. At the same time, the production of quantitative data for models where none was available (e.g., viscous effects on detonation stability) should be of immediate practical use.