The principal investigator proposes to study stability of compressible flows in ``real'' media featuring often-neglected effects such as viscosity, heat conduction, electromagnetic dynamics, phase-transition, non-thermoequilibrium, and chemical reaction, in the physically interesting (usually large-amplitude) regime where transition to instability may be expected to occur: for example, multidimensional stability of strong shock and detonation waves, or of classical shear flows. This involves interesting and nonstandard issues in singular perturbation theory, dynamical systems and bifurcation, spectral theory of linear operators, and nonlinear partial differential equations, and should result in the development of new mathematical tools of general application. The ultimate physical goal is an understanding of stability phenomena that is both more complete and more precise than can be obtained within simplified models: on the one hand resolving philosophical puzzles at the level of mathematical foundations and on the other yielding quantitative predictions at the level of practical application. The plan of attack centers around Evans function and related spectral techniques developed recently in the study of stability of viscous shock fronts.
The stability of regular flow patterns is an old and central topic in fluid, gas, and plasma dynamics, deciding which (stable) patterns will typically be observed, and which (unstable) are only mathematical and not physically observable solutions. The transition from stability to instability is of particular importance, since it usually signals the arising of alternative, more complicated flow patterns close to the original (now unstable) one- this is a way to understand complicated flows by the study of simpler and better-understood ones. Despite a large and well-known body of theory on this subject, dating back to the late 1800's, there are still many aspects that are poorly understood, particularly for compressive, viscous, or reacting flows. Here, we propose to study several of these issues arising in compressible gas and plasma dynamics, and in combustion, applications in which such usually neglected effects are of considerable practical importance. Our goal is, by including these mathematically problematic terms, to move existing theory from the qualitative to the quantitative regime, obtaining new information of use to practitioners at the same time that we advance the mathematical theory. The planned activities have both analytic and numerical components, and involve collaboration with domestic and foreign colleagues and with current and former graduate students and post doctorates. This may be expected to strengthen and extend existing networks of cooperation across field and institution, and to aid in training of graduate and postdoctoral students. The ultimate aim of these investigations, of quantitative predictions of transition to instability, would, if achieved, be of direct and practical use at the level of engineering, in chemical, manufacturing, and other processes.
