The principal investigator proposes to study a variety of problems in stability and behavior of shock and boundary layer type solutions of the equations of compressible gas dynamics, with an emphasis on global (i.e., amplitude-independent) analysis; new functional-analytic tools suitable for treatment of time-periodic, discrete, and or kinetic waves; and practical numerical testing of stability and bifurcation, especially for the large systems resulting from inclusion of viscous heat- and magnetic- or electromagnetic-inductive, phase-transitional, and reactive effects, or by Fourier transform/truncation in transverse modes of genuinely multidimensional solutions such as nonplanar (i.e., varying in transverse, or nonaxial, coordinates) flow in a cylindrical duct. The latter is expected to provide quantitative information of interest to physical practitioners in shock and detonation theory. A larger goal is to move beyond simple stability analysis to the study of nontrivial dynamics including bifurcation, interaction, and behavior of complex flows. The project 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 plan of attack centers around Evans function and related spectral techniques developed recently by the investigator and various collaborators.
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 appearance 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. Our goal is 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 expected to strengthen and extend existing networks of cooperation across field, 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.
, in particular achieving the primary goal of moving from articial problems and numerical feasibility studies to realistic physical systems and large-scale parallel computations. He in addition achieved unexpected breakthroughs in pattern formation/modulation of periodic waves. In particular, we rigorously established stability of large amplitude isentroic Navier-Stokes shocks and nonlinear stability of spectrally stable periodic Kuramoto Sivashinsky waves, two 30+ year open problems in quite different areas. The conservation and balance laws arising in compressible gas dynamics, elasticity, MHD, and detonation problems involve acoustic modes for which the dominant large-time effect is convection along characteristics. In the context of a stationary front or pulse solution, this translates to the property that the linearized operator about the wave has essential spectrum tangent at the origin to the imaginary axis, hence no spectral gap. Consequently, standard theorems on decomposition into center, stable, and unstable manifolds do not apply, and such basic dynamic principles had not been established and indeed were not clear to exist until the work of the PI in this grant period. In addition, 6 students completed a PhD; . 2 received IU dissertation year fellowships in arts and sciences, 1 an NSF postdoc; 3 are continuing. Zumbrun mentored NSF postdoc M. Johnson (2009-2011) andpostdoc A. Pogan (2010-2014), now assistant professors. Zumbrun presented a set of 2009 IMA summer school lectures, now a text in revision, and has developed a set of grad course notes on ODE. Former students Humpherys (2002) and Lyng (2002) both received NSF CAREER grants. With Barker and Humpherys, Zumbrun has developed an efficient and robust MATLAB-based package, STABLAB, for numerical stability evaluation. From 2009-2014 Zumbrun served as Chair.
