Kevin Zumbrun
Mathematics Department
Indiana University

Abstract: Transition to instability of detonation waves is typified by ``pulsating'', ``spinning'', or ``cellular'' behavior, appearing as bifurcation to time-periodic perturbations of the background wave. We discuss the rigorous characterization of this phenomenon within the context of the reactive Navier-Stokes equations, and the possibility of corresponding behavior in magnetohydrodynamic or phase-transitional shock waves. The main mathematical issues from the point of view of abstract bifurcation theory are the absence of a spectral gap between bifurcating modes and essential spectrum of the linearized operator about the wave and apparent loss of derivatives due to the incomplete parabolic smoothing (no mass diffusion) of the Navier-Stokes equations.