Large time behavior of fronts governed by eikonal equations

Abstract

Motivated by a model of solid combustion in heterogeneous media, we investigate the timeasymptotic 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).

Homogenization of First-Order Equations with $$(u/\varepsilon)$$ -Periodic Hamiltonians. Part I: Local Equations

“…See also Berestycki, Hamel [12], where pulsating fronts are studied in periodic media, these pulsating fronts being very similar to our corrector solutions. Finally, let us mention that the construction of correctors is related to the long time behaviour of solutions to Hamilton-Jacobi equations; such a problem has been studied by many authors, with seminal works of Fathi [22] and Roquejoffre [34] -see also Barles and Roquejoffre [8] and Barles and Souganidis [6] and references therein.…”

Homogenization results for a deterministic multi-domains periodic control problem

“…In [22], motivated by a model of solid combustion in heterogeneous media, Roquejoffre and the first author studied the time-asymptotic behaviour of flame fronts evolving with a periodic space-dependent normal velocity using the "level-sets approach".…”

“…In [22], results on this propagation are given and in particular on the asymptotic velocity but only in the case of Lipschitz continuous functions R. However an interesting case (which is the purpose of an entire (but formal) section in [22]) was when R is given in R 2 by…”

Surface Evolution Equations

“…Even if a = 0, the case when c or β depends on spatial variables is difficult to treat. If c/β depends only x periodically and independent of n and t, a large time behaviour is a recent topic as for example studied by G. Barles and J.-M. Roquejoffre (2003).…”