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

凯文Zumbrun提出攻击的冲击波和爆轰波的稳定性和行为的选择关键的开放问题，并在薄膜流，光学和各种其他情况下产生的周期性图案。 这些问题具有计算复杂性和多个长度和时间尺度上过程之间微妙的相互作用的特征，沿着的事实是，它们涉及基本的和经常发生的物理现象，已经在几十年的时间内进行了大量的研究，但在严格的数学水平上仍然没有得到解决。重要的是要注意，这是一个例子，数学不只是在逻辑上验证已经观察到的物理或实验原理，但在当前的数值和实验不足以解决的设置中找到秩序。 几个计划中的子项目涉及数字辅助证明使用科学计算与保证误差范围。 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，并在反应或电离流的微妙情况下系统地探索气体和流体动力学的物理行为。 大多数建议的计算，特别是那些涉及多个维度和粘性效应同时进行，从来没有成功地进行过-因此有一个实质性的数值/计算组件，这个程序以及解决的问题涉及有趣的和非标准的问题，在转折点理论，谱理论的nonselfadjoint运营商，非线性偏微分方程，和图案的形成。所考虑的问题是长期存在的基本物理利益，其解决方案将需要显着的新工具。特别是，严格的数值稳定性验证算法的发展;无界域上的解析系数转折点问题的治疗，特别是在无穷大的转折点;和惊人的粘度与高活化能相结合的影响的调查似乎可能是转型的稳定性和分叉的流体和气体动力流的研究。这些问题中的每一个都涉及多个尺度（刚度）和没有频谱间隙的技术困难;它们的成功分析涉及在线性水平上通过固定相位和相关的复杂分析方法以及在非线性水平上通过PI和合作者开发的相位提取/调制技术来进行精细消除。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的潜力。 与此同时，为没有可用模型的模型提供定量数据（例如，粘性对爆轰稳定性的影响）应该立即得到实际应用。

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.