The Equations of One-Dimensional Unsteady Flame Propagation: Existence and Uniqueness

This paper deals with the mathematical analysis of a system of partial differential equations describing the time-dependent propagation of a planar flame front within the framework of the well-known isobaric approximation of slow combustion. The problem to be investigated takes the form of a nonlinear mixed initial-boundary value problem in an infinite one-dimensional domain. We show the existence and uniqueness of weak and classical solutions of this problem, depending on the assumptions on the initial data and on the nonlinear temperature dependence of the chemical reaction rates. The crucial point lies in the introduction of a Lagrangian space coordinate, which uncouples the reaction-diffusion equations for the combustion variables from the remaining hydrodynamical subsystem. The analysis then uses some classical arguments of functional analysis, such as the application of the theory of linear semigroups to nonlinear partial differential equations.