Motivated by a model of solid combustion in heterogeneous media, we investigate the time-asymptotic behaviour of flame fronts evolving with a periodic space-dependent normal velocity; using the so-called “level set approach” we are led to study the large time behaviour of solutions of eikonal equations. We first provide a general approach which shows that the asymptotic normal velocity of such a flame front depends only on its normal direction and is given by the homogenized Hamiltonian of the eikonal equation. Then we turn to a more precise study of the asymptotic behaviour of the flame front when the initial front is a graph of a periodic function: in this case, the front moves asymptotically with a constant normal velocity and we are able to prove that, in coordinates moving with this constant velocity, the front has a time-periodic asymptotic behaviour in the following two cases: (i) when there is a straight line of maximal speed, and (ii) when the space dimension is 2. These results are obtained by using homogenization, control theory and dynamical systems methods (Aubry–Mather sets).
Classification :
46-XX, 60-XX
Mots-clés :
Front propagation; level set approach; eikonal equations; periodic solutions; homogenization; Aubry--Mather set
Affiliations des auteurs :
Guy Barles
1
;
Jean-Michel Roquejoffre
2
1
Université de Tours, France
2
Université Paul Sabatier, Toulouse, France
Guy Barles; Jean-Michel Roquejoffre. Large time behavior of fronts governed by eikonal equations. Interfaces and free boundaries, Tome 5 (2003) no. 1, pp. 83-102. doi: 10.4171/ifb/73
@article{10_4171_ifb_73,
author = {Guy Barles and Jean-Michel Roquejoffre},
title = {Large time behavior of fronts governed by eikonal equations},
journal = {Interfaces and free boundaries},
pages = {83--102},
year = {2003},
volume = {5},
number = {1},
doi = {10.4171/ifb/73},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/73/}
}
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AU - Guy Barles
AU - Jean-Michel Roquejoffre
TI - Large time behavior of fronts governed by eikonal equations
JO - Interfaces and free boundaries
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