Asymptotic Analysis - Volume 88, issue 1-2

Authors: Kozlowski, K.K.

Article Type: Research Article

Abstract: By using Riemann–Hilbert problem based techniques, we obtain the asymptotic expansion of lacunary Toeplitz determinants det N [cℓa −mb [f]] generated by holomorphic symbols, where ℓa =a (resp. mb =b) except for a finite subset of indices a=h1 ,…,hn (resp. b=t1 ,…,tr ). In addition to the usual Szegö asymptotics, our answer involves a determinant of size n+r.

Keywords: Toeplitz determinants, Toeplitz minors, large-size determinants

DOI: 10.3233/ASY-131210

Citation: Asymptotic Analysis, vol. 88, no. 1-2, pp. 1-16, 2014

Authors: Farwig, Reinhard | Schulz, Raphael | Yamazaki, Masao

Article Type: Research Article

Abstract: We study in the whole space Rn the behaviour of solutions to the Boussinesq equations at large distances. Therefore, we investigate the solvability of these equations in weighted L∞ -spaces and determine the asymptotic profile for sufficiently fast decaying initial data. For n=2,3 we are able to construct initial data such that the velocity exhibits an interesting concentration–diffusion phenomenon.

Keywords: instationary Boussinesq equations, rate of decay in space, mild and strong solutions, weighted spaces, concentration–diffusion

DOI: 10.3233/ASY-131211

Citation: Asymptotic Analysis, vol. 88, no. 1-2, pp. 17-41, 2014

Authors: Chesnel, Lucas | Claeys, Xavier | Nazarov, Sergey A.

Article Type: Research Article

Abstract: We study a 2D scalar harmonic wave transmission problem between a classical dielectric and a medium with a real-valued negative permittivity/permeability which models a metal at optical frequency or an ideal negative metamaterial. We highlight an unusual instability phenomenon for this problem when the interface between the two media presents a rounded corner. To establish this result, we provide an asymptotic expansion of the solution, when it is well-defined, in the geometry with a rounded corner. Then, we prove error estimates. Finally, a careful study of the asymptotic expansion allows us to conclude that the solution, when it is well-defined, …depends critically on the value of the rounding parameter. We end the paper with a numerical illustration of this instability phenomenon. Show more

Keywords: negative materials, corner, plasmonic, metamaterial, sign-changing coefficients

DOI: 10.3233/ASY-141214

Citation: Asymptotic Analysis, vol. 88, no. 1-2, pp. 43-74, 2014

Authors: Ono, Kosuke

Article Type: Research Article

Abstract: We consider the Cauchy problem for mildly degenerate Kirchhoff type dissipative wave equations: ρu″+‖A1/2 u(t)‖2γ Au+u′=0 in RN ×R+ , u(x,0)=u0 (x), u′(x,0)=u1 (x) in RN , with ρ>0, γ≥1, and ‖A1/2 u0 ‖>0. When either the coefficient ρ or the initial data {u0 ,u1 } are small, we prove the existence of global solutions by using several identities for energies. Moreover, we derive lower decay estimates of the solutions, and upper decay estimates of their second order derivatives.

Keywords: Kirchhoff strings, degenerate equations, decay rates

DOI: 10.3233/ASY-141215

Citation: Asymptotic Analysis, vol. 88, no. 1-2, pp. 75-92, 2014

Authors: Nasreddine, Elissar

Article Type: Research Article

Abstract: We consider a model of individual clustering with two specific reproduction rates and small diffusion parameter in one space dimension. It consists of a drift–diffusion equation for the population density coupled to an elliptic equation for the velocity of individuals. We prove the convergence (in suitable topologies) of the solution of the problem to the unique solution of the limit transport problem, as the diffusion coefficient tends to zero.

Keywords: elliptic system, parabolic equation, vanishing diffusion, nonlocal transport problem, existence and uniqueness of smooth solution, a priori estimates, compactness method

DOI: 10.3233/ASY-141218

Citation: Asymptotic Analysis, vol. 88, no. 1-2, pp. 93-110, 2014

Authors: Ou, Chunhua | Wong, R.

Article Type: Research Article

Abstract: In this paper we re-visit the Lagerstrom problem y" + (n-1)/r y' + εyy' = 0, y(1) = 0, y(∞) = 1, where ε is a small positive real number and n is a positive integer (or any real number greater than 2). Using rigorous analysis, a generalized asymptotic expansion, as ε→0, is derived for the solution of this problem. A trans-series expansion of the solution for large values of r is also presented; the leading term coefficient is determined by a connection formula between the values of the solution at the two points r=1 and r=∞. An extension …and a discussion of the problem for n∈[1,2) is also given. Show more

Keywords: boundary value problem, multiple scale, singular perturbation

DOI: 10.3233/ASY-131212

Citation: Asymptotic Analysis, vol. 88, no. 1-2, pp. 111-123, 2014