Asymptotic Analysis - Volume 10, issue 2

Authors: Robbiano, L.

Article Type: Research Article

Abstract: Dans cet article, on donne une estimation de l'énergie du contrôle de la solution d'un problème hyperbolique. Cette estimation est de la forme eC/ε2 si ε est le maximum de la norme de la solution à l'instant T.

DOI: 10.3233/ASY-1995-10201

Citation: Asymptotic Analysis, vol. 10, no. 2, pp. 95-115, 1995

Authors: Corli, Andrea

Article Type: Research Article

Abstract: We consider in this paper a strictly hyperbolic system of conservation laws, in one space dimension. We suppose that a shock wave solution to this system is given and superimpose to it small amplitude fast oscillations. These oscillations are described by sets of phase functions, and resonances are taken into account. We justify the asymptotic expansions given by the weakly non-linear geometric optics: the oscillating part of the perturbed solution is approximated by an almost periodic function (a profile) which is solution to a non-linear integro-differential mixed problem. We give at the same time an asymptotics to the shock front.

DOI: 10.3233/ASY-1995-10202

Citation: Asymptotic Analysis, vol. 10, no. 2, pp. 117-172, 1995

Authors: Nochetto, R.H. | Paolini, M. | Verdi, C.

Article Type: Research Article

Abstract: A singularly perturbed double obstacle problem is examined as a variational tool for the approximation of the geometric motion of fronts. The relaxation parameter is space-time dependent, thereby allowing the control of transition layer thickness and related interface pointwise accuracy. Optimal order interface error estimates are derived for smooth evolutions. The estimates have a local character for small time, namely they depend on the relaxation parameter local magnitude. The proof is based on constructing suitable sub and supersolutions, which incorporate a number of shape corrections to the basic standing wave profile, and using a modified distance function to the front. …Numerical simulations illustrate how the variable transition layer thickness can be exploited in dealing with large curvatures and ultimately in resolving singularities. Show more

DOI: 10.3233/ASY-1995-10203

Citation: Asymptotic Analysis, vol. 10, no. 2, pp. 173-198, 1995