Asymptotic Analysis - Volume 83, issue 3

Authors: De Maio, U. | Nandakumaran, A.K.

Article Type: Research Article

Abstract: In this paper, by using the Hilbert Uniqueness Method (HUM), we study the exact controllability problem described by the wave equation in a three-dimensional horizontal domain bounded at the bottom by a smooth wall and at the top by a rough wall. The latter is assumed to consist in a plane wall covered with periodically distributed asperities whose size depends on a small parameter ε>0, and with a fixed height. Our aim is to obtain the exact controllability for the homogenized equation. In the process, we study the asymptotic analysis of wave equation in two setups, namely solution by standard …weak formulation and solution by transposition method. Show more

Keywords: wave equation, homogenization, oscillating boundary, exact controllability

DOI: 10.3233/ASY-2012-1153

Citation: Asymptotic Analysis, vol. 83, no. 3, pp. 189-206, 2013

Authors: Prange, Christophe

Article Type: Research Article

Abstract: This paper is concerned with the homogenization of the Dirichlet eigenvalue problem, posed in a bounded domain Ω⊂R2 , for a vectorial elliptic operator −∇·Aε (·)∇ with ε-periodic coefficients. We analyse the asymptotics of the eigenvalues λε,k when ε→0, the mode k being fixed. A first-order asymptotic expansion is proved for λε,k in the case when Ω is either a smooth uniformly convex domain, or a convex polygonal domain with sides of slopes satisfying a small divisors assumption. Our results extend those of Moskow and Vogelius in Proc. Roy. Soc. Edinburgh Sect. A 127(6) (1997), 1263–1299 restricted to …scalar operators and convex polygonal domains with sides of rational slopes. We take advantage of the recent progress due to Gérard-Varet and Masmoudi [J. Eur. Math. Soc. 13 (2011), 1477–1503; Acta Math. 209 (2012), 133–178] in the homogenization of boundary layer type systems. Show more

Keywords: homogenization, boundary layers, elliptic systems, regularity estimates in irregular domains, low frequency waves

DOI: 10.3233/ASY-121158

Citation: Asymptotic Analysis, vol. 83, no. 3, pp. 207-235, 2013

Authors: Klovienė, N. | Pileckas, K.

Article Type: Research Article

Abstract: The non-stationary equations describing the motion of the second grade fluid are studied in an infinite three-dimensional pipe of arbitrary cross-section. For sufficiently small data the existence of the unique Poiseuille type solution having a given time-dependent flow rate (flux) is proved. The velocity field U has all three components. However, we show that components U1 , U2 are secondary in comparison with the axial velocity U3 .

Keywords: Poiseuille solution, pipe flow, non-stationary solutions, second grade fluid flow

DOI: 10.3233/ASY-121159

Citation: Asymptotic Analysis, vol. 83, no. 3, pp. 237-262, 2013

Authors: Ammari, Kaïs | Nicaise, Serge | Pignotti, Cristina

Article Type: Research Article

Abstract: In this paper we consider some stabilization problems for the wave equation with switching time-delay. We prove exponential stability results for appropriate damping coefficients. The proof of the main results is based on D'Alembert formula, observability inequality and some energy estimates. More general problems, like the Petrovsky system, are also discussed.

Keywords: stabilization with time-delay, pointwise stabilization, boundary/internal stabilization, switching control, wave equations

DOI: 10.3233/ASY-131163

Citation: Asymptotic Analysis, vol. 83, no. 3, pp. 263-283, 2013