Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames

LEDERMAN, CLAUDIA; ROQUEJOFFRE, JEAN-MICHEL; WOLANSKI, NOEMÍ

ANNALI DI MATEMATICA PURA ED APPLICATA

Springer-Verlag

Lugar: Berlin; Año: 2004 vol. 183 p. 173 - 239

0373-3114

This paper is devoted to the justification of an asymptotic model for quasisteady three-dimensional spherical flames proposed by G. Joulin [17]. The paper [17] derives, by means of a three-scale matched asymptotics, starting from the classical thermo-diffusive model with high activation energies, an integro-differential equation for the flame radius. In the derivation, it is essential for the Lewis Number  -i.e. the ratio between thermal and molecular diffusion- to be strictly less than unity. If {epsilon} is the inverse of the -reduced- activation energy, the idea underlying the construction of [17] is that (i) the time scale of the radius motion is {epsilon}-2, and that (ii) at each time step, the solution is {epsilon}-close to a steady solution. In this paper, we give a rigorous proof of the validity of this model under the restriction that the Lewis number is close to 1 -independently of the order of magnitude of the activation energy. The method used comprises three steps: (i) a linear stability analysis near a steady -or quasi-steady- solution, which justifies the fact that the relevant time scale is {epsilon}-2; (ii) the rigorous construction of an approximate solution; (iii) a nonlinear stability argument.