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The journal Asymptotic Analysis fulfills a twofold function. It aims at publishing original mathematical results in the asymptotic theory of problems affected by the presence of small or large parameters on the one hand, and at giving specific indications of their possible applications to different fields of natural sciences on the other hand.
Asymptotic Analysis thus provides mathematicians with a concentrated source of newly acquired information which they may need in the analysis of asymptotic problems.
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
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