Asymptotic Analysis - Volume 54, issue 3-4

Authors: Kachmar, Ayman

Article Type: Research Article

Abstract: We study a generalized Ginzburg–Landau equation that models a sample formed of a superconducting/normal junction and which is not submitted to an applied magnetic field. We prove the existence of a unique positive (and bounded) solution of this equation. In the particular case when the domain is the entire plane, we determine the explicit expression of the solution (and we find that it satisfies a Robin (de Gennes) boundary condition on the boundary of the superconducting side). Using the result of the entire plane, we determine for the case of general domains, the asymptotic behavior of the solution for large …values of the Ginzburg–Landau parameter. The main tools are Hopf's Lemma, the Strong Maximum Principle, elliptic estimates and Agmon type estimates. Show more

Keywords: generalized Ginzburg–Landau energy functional, proximity effects, global minimizers, unique positive solution

Citation: Asymptotic Analysis, vol. 54, no. 3-4, pp. 125-164, 2007

Authors: Wang, X.S. | Wong, R.

Article Type: Research Article

Abstract: An asymptotic formula is derived for the sum In (1|q):=Σn k=0 fn (k)qgn (k) as n→∞, where fn (k) and gn (k) are functions defined on nonnegative integers and 0<q<1. This formula is a discrete analogue of Laplace's approximation for integrals. Corresponding results are also provided for the more general sum In (z|q):=Σn k=0 fn (k)qgn (k) zk which is typically an nth order polynomial. The results obtained are then used to give asymptotic formulas for the q−1 -Hermite polynomial hn (x|q), the Stieltjes–Wigert polynomial Sn (x; q) and the q-Laguerre polynomial Lα n …(x; q). Show more

Keywords: Laplace's approximation, q-Airy function, q^−1-Hermite polynomial, Stieltjes–Wigert polynomial, q-Laguerre polynomial

Citation: Asymptotic Analysis, vol. 54, no. 3-4, pp. 165-180, 2007

Authors: Sabu, N.

Article Type: Research Article

Abstract: In this paper, we consider the boundary value problem for thin piezoelectric shells with variable thickness subjected only to mechanical body force and we show that, as the thickness of the shell goes to zero, the solution of the three-dimensional equations converge to the solution of two-dimensional problem.

Citation: Asymptotic Analysis, vol. 54, no. 3-4, pp. 181-196, 2007

Authors: Lu, Songsong

Article Type: Research Article

Abstract: The long time behavior of the solutions of a general reaction–diffusion system (RDS) that covers many examples, such as the RDS with polynomial nonlinearity and Ginzburg–Landau equation, is discussed. First, the existence of a compact uniform attractor 𝒜0 in H is proved without additional assumptions on the interaction functions. Then the structure of the attractor is obtained for a certain class of interaction functions without strong translation compactness. For instance, the interaction functions are not required to be uniformly continuous. Moreover, an interesting problem arises naturally from this paper.

Keywords: uniform attractor, reaction–diffusion system, normal symbol

Citation: Asymptotic Analysis, vol. 54, no. 3-4, pp. 197-210, 2007

Authors: Pileckas, K. | Zaleskis, L.

Article Type: Research Article

Abstract: The Stokes problem is studied in the domain Ω⊂R3 coinciding outside the ball BR ={x∈R3 : |x|<R} with the parabolically growing layer Ω={x∈R3 : x′ =(x1 , x2 )∈R2 , |x3 |<h(x′ )}, where h(x′ ) is a smooth function, h(x′ )≥h0 >0 ∀x′ ∈R2 and h(x′ )=|x′ |β ≡rβ , β∈(0, 1), for r>1. Coercive estimates of the solution to the Stokes problem are proved in a scale of weighted function spaces with the norm determined by a stepwise anisotropic distribution of weight factors.

Keywords: stationary Stokes problem, unbounded domains, coercive estimates, weighted function spaces

Citation: Asymptotic Analysis, vol. 54, no. 3-4, pp. 211-233, 2007

Authors: Desvillettes, Laurent | Mouhot, Clément

Article Type: Research Article

Abstract: We consider the spatially homogeneous Boltzmann equation for regularized soft potentials and Grad's angular cutoff. We prove that uniform (in time) bounds in L1 ((1+|v|s ) dv) and Hk norms, s, k≥0 hold for its solution. The proof is based on the mixture of estimates of polynomial growth in time of those norms together with the quantitative results of relaxation to equilibrium in L1 obtained by the so-called “entropy–entropy production” method in the context of dissipative systems with slowly growing a priori bounds (G. Toscani and C. Villani, J. Statist. Phys. 98 (2000), 1279–1309).

Keywords: Boltzmann equation, spatially homogeneous, soft potentials, moment bounds, regularity bounds, uniform in time

Citation: Asymptotic Analysis, vol. 54, no. 3-4, pp. 235-245, 2007

Article Type: Other

Citation: Asymptotic Analysis, vol. 54, no. 3-4, pp. 247-248, 2007