Asymptotic Analysis - Volume 97, issue 3-4

Authors: Dai, Mimi | Feireisl, Eduard | Rocca, Elisabetta | Schimperna, Giulio | Schonbek, Maria E.

Article Type: Research Article

Abstract: We study a PDE system describing the motion of liquid crystals by means of the Q -tensor description for the crystals coupled with the incompressible Navier–Stokes system. Using the method of Fourier splitting, we show that solutions of the system tend to the isotropic state at the rate ( 1 + t ) − 3 / 2 as t → ∞ .

Keywords: liquid crystal, Q-tensor description, long-time behavior, Fourier splitting

DOI: 10.3233/ASY-151348

Citation: Asymptotic Analysis, vol. 97, no. 3-4, pp. 189-210, 2016

Authors: Delourme, Bérangère | Schmidt, Kersten | Semin, Adrien

Article Type: Research Article

Abstract: The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium. We provide and justify a high order asymptotic expansion which takes into account the boundary layer effect occurring in the vicinity of the periodic layer as well as the corner singularities appearing in the neighborhood of the extremities of the layer. Our approach combines the method of matched asymptotic expansions and the method of periodic surface homogenization.

Keywords: asymptotic analysis, periodic surface homogenization, singular asymptotic expansions

DOI: 10.3233/ASY-151350

Citation: Asymptotic Analysis, vol. 97, no. 3-4, pp. 211-264, 2016

Authors: Klevtsovskiy, A.V. | Mel’nyk, T.A.

Article Type: Research Article

Abstract: A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin domain Ω ε coinciding with two thin rectangles connected through a joint of diameter O ( ε ) . A rigorous procedure is developed to construct the complete asymptotic expansion for the solution as the small parameter ε → 0 . Energetic and uniform pointwise estimates for the difference between the solution of the starting problem (ε > 0 ) and the solution of the corresponding limit problem (ε = 0 ) …are proved, from which the influence of the geometric irregularity of the joint is observed. Show more

Keywords: asymptotic expansion, asymptotic estimate, thin domain with a local geometrical irregularity

DOI: 10.3233/ASY-151352

Citation: Asymptotic Analysis, vol. 97, no. 3-4, pp. 265-290, 2016

Authors: Matei, Basarab

Article Type: Research Article

Abstract: In this paper we are concerned with the reconstruction of a class of measures on the square from the sampling of its Fourier coefficients on some sparse set of points. We show that the exact reconstruction of a weighted Dirac sum measure is still possible when one knows a finite number of non-adaptive linear measurements of the spectrum. Surprisingly, these measurements are defined on a model set, i.e. quasicrystal.

Keywords: Fourier analysis, irregular sampling theory

DOI: 10.3233/ASY-151353

Citation: Asymptotic Analysis, vol. 97, no. 3-4, pp. 291-299, 2016

Authors: Mohammed, Mogtaba | Sango, Mamadou

Article Type: Research Article

Abstract: In this paper, we investigate a linear hyperbolic stochastic partial differential equation (SPDE) with rapidly oscillating ϵ -periodic coefficients in a domain with small holes (of size-ϵ ) under Neumann conditions on the boundary of the holes and Dirichlet condition on the exterior boundary. When the number of these holes approach infinity, i.e. their sizes approach zero, the homogenized problem is a hyperbolic SPDE with constant coefficients in the domain without perforations. Moreover the convergence of the associated energy to that of the homogenized system is established.

Keywords: homogenization, hyperbolic SPDEs, Neumann problem, perforated domains, probabilistic compactness results

DOI: 10.3233/ASY-151355

Citation: Asymptotic Analysis, vol. 97, no. 3-4, pp. 301-327, 2016

Authors: Babich, P.V. | Levenshtam, V.B.

Article Type: Research Article

Abstract: This paper is devoted to the first boundary value problem for the heat equation with a fast oscillating source. Direct and inverse problems are solved. The direct problem is to construct and justify an asymptotic expansion for the solution under appropriate assumptions. The inverse problem is to find the source if the value of two-term asymptotic expansion for solution at some point of space is given.

Keywords: asymptotics, heat equation, high-frequency, inverse problem

DOI: 10.3233/ASY-161356

Citation: Asymptotic Analysis, vol. 97, no. 3-4, pp. 329-336, 2016

Authors: Perjan, Andrei | Rusu, Galina

Article Type: Research Article

Abstract: In a real Hilbert space H we consider the following singularly perturbed Cauchy problem ( P ε δ ) ε u ε δ ″ ( t ) + δ u ε δ ′ ( t ) + A u ε δ ( t ) = f ( t ) , t ∈ ( 0 , T ) , u ε δ ( 0 ) = u 0 , u ε …δ ′ ( 0 ) = u 1 , where u 0 , u 1 ∈ H , f : [ 0 , T ] ↦ H and ε , δ are two small parameters. We study the behavior of the solutions u ε δ to the problem (P ε δ ) in two different cases: (i) when ε → 0 and δ ⩾ δ 0 > 0 ; (ii) when ε → 0 and δ → 0 . We obtain a priori estimates of the solutions to the perturbed problem, which are uniform with respect to the parameters, and a relationship between the solutions to both problems. We establish that the solution to the unperturbed problem has a singular behavior with respect to the parameters in the neighborhood of t = 0 . We describe the boundary layer and the boundary layer function in both cases. when ε → 0 and δ ⩾ δ 0 > 0 ; when ε → 0 and δ → 0 . Show more

Keywords: singular perturbation, abstract second order Cauchy problem, boundary layer function, a priori estimate

DOI: 10.3233/ASY-161357

Citation: Asymptotic Analysis, vol. 97, no. 3-4, pp. 337-349, 2016

Authors: Hamza, M.A.

Article Type: Research Article

Abstract: We consider in this paper a large class of perturbed semilinear wave equations with critical (in the conformal transform sense) power nonlinearity. We will show that the blow-up rate of any singular solution is given by the solution of the non-perturbed associated ODE. The result in the radial case has been proved in Math. Phys. Anal. Geom. 18 (1) (2015), Art. 15. The same approach will be followed here, but the main difference is to construct a Lyapunov functional in similarity variables valid in the non-radial case, which is far from being trivial. That functional is obtained by combining …some classical estimates and a new identity of the Pohozaev type obtained by multiplying Eq. (1.7) by y · ∇ w in a suitable weighted space. Show more

Keywords: semilinear wave equation, blow-up, perturbations, conformal exponent, Pohozaev identity

DOI: 10.3233/ASY-161358

Citation: Asymptotic Analysis, vol. 97, no. 3-4, pp. 351-378, 2016

Article Type: Other

Citation: Asymptotic Analysis, vol. 97, no. 3-4, pp. 379-380, 2016