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

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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.