Asymptotic Analysis - Volume 14, issue 3

Authors: Saint Jean Paulin, Jeannine | Tcheugoué Tébou, Louis Roder

Article Type: Research Article

Abstract: We consider the internal controllability of a generalized wave equation in a periodically perforated domain, with a Fourier type boundary condition on the boundary of the holes. First, we establish by the Hilbert uniqueness method, HUM, introduced by J.L. Lions, the existence of an exact control; afterwards, we prove that the sequence of controls weakly converges to a function which is the exact control of the homogenized system. We also prove a strong convergence result for the sequence of controls.

DOI: 10.3233/ASY-1997-14301

Citation: Asymptotic Analysis, vol. 14, no. 3, pp. 193-221, 1997

Authors: Nazarov, S.A. | Specovius-Neugebauer, M.

Article Type: Research Article

Abstract: Let Ω⊂R3 be an exterior domain and u a solution of the Dirichlet problem for the Stokes system. Let {GR } be a set of bounded domains which contain ∂Ω for any R≥1 and exhaust Ω as R→∞. The problem is investigated how u can be approximated by solutions uR of boundary problems which are defined on the bounded subdomain Ω∩GR . On the external boundary ∂GR an artificial boundary condition BuR =h has to be added. In the three cases – uR = 0, TuR ·ν=0 and TuR ·ν+A·u=0 on ∂GR – formal asymptotic …estimates are derived for u - uR . It turns out that the mixed boundary condition leads to the best asymptotic decay if the matrix A(x) is chosen in a proper way. For this boundary condition the unique solvability and R-independent estimates are proved in weighted Sobolev spaces. With these results the formal error estimates are justified rigorously. Show more

DOI: 10.3233/ASY-1997-14302

Citation: Asymptotic Analysis, vol. 14, no. 3, pp. 223-255, 1997

Authors: McLean, W. | Thomée, V.

Article Type: Research Article

Abstract: We study the exponential decay of discrete and continuous solutions of a Volterra type integro-differential equation, in which the integral operator is a convolution of an exponentially decreasing scalar positive definite kernel and a positive definite operator, such as an elliptic differential operator. The equation is discretized in time by the backward Euler method in combination with convolution quadrature.

Keywords: Partial integro-differential equation, convolution quadrature

DOI: 10.3233/ASY-1997-14303

Citation: Asymptotic Analysis, vol. 14, no. 3, pp. 257-276, 1997

Authors: Barles, G. | Georgelin, C. | Souganidis, P.E.

Article Type: Research Article

Abstract: In this paper we study the rigorous connections between reaction–diffusion equations of ZFK-type, which arise as simple models in combustion theory, and the propagating fronts, i.e., propagating flames, they generate for large times. Our main result is that the associated fronts propagate globally in time with normal velocity equal to the speed of the stable travelling wave associated with the reaction–diffusion equation. We point out that this result does not follow from any parts of the, by now, classical methods to study front propagation.

DOI: 10.3233/ASY-1997-14304

Citation: Asymptotic Analysis, vol. 14, no. 3, pp. 277-292, 1997