IUMJ

Title: Heteroclinic orbits with fast transitions: a new construction of detonation profiles

Authors: Mark Williams

Issue: Volume 59 (2010), Issue 3, 1145-1210

Abstract:

We give a direct and elementary construction of strong detonation profiles for the reactive Navier-Stokes equations (RNS), starting with an inviscid template given by a solution to the Zeldovitch-von Neumann-Doering (ZND) equations. Assuming that the viscosity, heat conductivity, and species diffusion coefficients in RNS are all proportional to $\epsilon$, we construct detonation profiles $w^{\epsilon}(x)$ that are exact solutions of RNS which converge in an appropriate sense to the given ZND profile as $\epsilon \to 0$. The construction is explicit in the sense that it produces an arbitrarily high order expansion in powers of $\epsilon$ for $w^{\epsilon}$, and the coefficients in the expansion satisfy simple, explicit ODEs, which are \emph{linear} except in the case of the leading term. Moreover, the leading "slow" term in the expansion is the original ZND profile, and the burned and unburned endstates of each RNS profile $w^{\epsilon}$ coincide with those of the given ZND profile.\par The method used here is applicable to a variety of singular perturbation problems in which one seeks to construct smooth ``viscous profiles', involving both slow and fast transition regions, that converge to discontinuous "inviscid profiles" as a viscosity parameter $\epsilon$ tends to $0$. The method is applicable, for example, to second-order systems that cannot be written in conservative form, and can be used to construct solutions with fast transitions in situations, like two-point boundary problems, where no rest points or higher dimensional invariant manifolds are present in the "reduced problem".