The principal investigator proposes several projects concerning multi-dimensional stability of flows in compressible, viscous, and reacting media. These include both shear flows of classical hydrodynamic stability and compressive flows of shock wave and combustion theory, the former exhibiting local symmetry parallel to and the latter normal to the flow. The unifying mathematical theme in these problems is the appearance of multiple length scales corresponding to small-scale transport and large-scale convective effects, with associated ``stiffness'' in the linearized perturbation problem. This leads to interesting, nonstandard issues in spectral and semigroup theory. At the same time, the inclusion of small-scale transport effects is highly desirable from the point of view of physical applications, which often occur at scales where these effects might be expected to be significant.
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 arisal 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.
