Pointwise Green function bounds and stability of combustion waves
Abstract:
In this talk, I describe the establishment of sharp pointwise Green function bounds and linearized and nonlinear stability for traveling wave solutions of an abstract viscous combustion model including both Majda's model and the full reacting compressible Navier-Stokes equations with artificial viscosity, under the necessary conditions of strong spectral stability, i.e., stable point spectrum of the linearized operator about the wave, transversality of the profile as a connection in the traveling-wave ODE, and hyperbolic stability of the associated Chapman-Jouguet (square-wave) approximation. These results apply to combustion waves of any type (weak or strong, detonations or deflagrations), and they reduce verification of stability to a numerically checkable Evans-function condition. Together with previous spectral results of Lyng and Zumbrun, these results give immediately stability of small-amplitude strong detonations in the small heat-release (i.e., fluid-dynamical) limit, simplifying and greatly extending previous results obtained by energy methods by Liu & Ying and Tesei &Tan for Majda's model and the reactive Navier--Stokes equations, respectively. This is joint work with M. Raoofi, B. Texier, and K. Zumbrun.
Applied and Interdisciplinary Mathematics Seminar