Asymptotic Analysis - Volume 52, issue 1-2

Authors: Nazarov, Sergueï A. | Videman, Juha H.

Article Type: Research Article

Abstract: We derive a nonlinear second-order differential equation for the pressure approximation in hydrodynamic lubrication. This equation, in contrast to the classical Reynolds equation, takes into account both the inertial and the curvature effects and its solution corresponds to the first two terms in the asymptotic pressure expansion. The equation is rigorously justified with optimal error estimates in parameter dependent Sobolev norms and in Hölder norms. It is also applied to the classical problem of journal bearing.

Keywords: Navier–Stokes equations, Reynolds equation, thin flows, lubrication

Citation: Asymptotic Analysis, vol. 52, no. 1-2, pp. 1-36, 2007

Authors: Ignat, Liviu I.

Article Type: Research Article

Abstract: We consider a semidiscrete scheme for the linear Schrödinger equation with high order dissipative term. We obtain maximum norm estimates for its solutions and we prove global Strichartz estimates for the considered model, estimates that are uniform with respect to the mesh size. The methods we employ are based on classical arguments of harmonic analysis.

Keywords: finite differences, Schrödinger equations, Strichartz estimates

Citation: Asymptotic Analysis, vol. 52, no. 1-2, pp. 37-51, 2007

Authors: Porretta, A.

Article Type: Research Article

Abstract: Given a nonnegative bounded Radon measure μ on Ω⊂RN , we discuss the existence or nonexistence of minima of infinite energy (so-called weak minima, T-minima, renormalized minima) for functionals like J(v)=∫Ω a(x,v)|∇v|p  dx−∫Ω v dμ, where p>1. In most of our results, a(x,s) is coercive. According to the behavior of $s\mapsto a(x,s)$ at infinity, existence or nonexistence of such minima is proved, and the convergence of approximating minima of regularized functionals is studied. Differences arise whether the measure charges or not sets of null p-capacity and/or a(x,s) blows-up at infinity. Lastly, some results are proved when a(x,s) degenerates at …infinity. Show more

Citation: Asymptotic Analysis, vol. 52, no. 1-2, pp. 53-94, 2007

Authors: Schlömerkemper, Anja

Article Type: Research Article

Abstract: Let ℒ be a lattice in $\mathbb{R}^{d}$ , d≥2, and let $A\subset\mathbb{R}^{d}$ be a Lipschitz domain which satisfies some additional weak technical regularity assumption. In the first part of the paper we consider certain lattice sums over points which are close to $\curpartial A$ . The main result is that these lattice sums approximate corresponding surface integrals for small lattice spacing. This is not obvious since the thickness of the domain of summation is comparable to the scale of the lattice. In the second part of the paper we study a specific singular lattice sum …in d≥2 and prove that this lattice sum converges as the lattice spacing tends to zero. This lattice sum and its convergence are of interest in lattice-to-continuum approximations in electromagnetic theories—as is the above approximation of surface integrals by lattice sums. This work generalizes previous results (Schlömerkemper, Arch. Rational Mech. Anal. 176 (2005), 227–269) from d=3 to d≥2 and to a more general geometric setting, which is no longer restricted to nested sets. Show more

Keywords: approximation of surface integrals, lattice-to-continuum theories, convergence of multidimensional singular series, magnetostatics

Citation: Asymptotic Analysis, vol. 52, no. 1-2, pp. 95-115, 2007

Authors: Moulahi, Ammar

Article Type: Research Article

Abstract: On étudie le problème de stabilisation des équations de Maxwell à l'extérieur d'un obstacle borné en dimension deux. On développe certains résultats concernant l'équation des ondes amorties de type Neumann et sous la condition de contrôle géométrique extérieur, on obtient le comportement des solutions pour des systèmes de Maxwell en grand temps.

Citation: Asymptotic Analysis, vol. 52, no. 1-2, pp. 117-141, 2007

Authors: Pazoto, Ademir F. | Rossi, Julio D.

Article Type: Research Article

Abstract: We study the semilinear nonlocal equation ut =J*u−u−up in the whole ${\mathbb{R}}^{N}$ . First, we prove the global well-posedness for initial conditions $u(x,0)=u_{0}(x)\in L^{1}({\mathbb{R}}^{N})\cap L^{\infty}({\mathbb{R}}^{N})$ . Next, we obtain the long time behaviour of the solutions. We show that different behaviours are possible depending on the exponent p and the kernel J: finite time extinction for p<1, faster than exponential decay for the linear case p=1, a weakly nonlinear behaviour for p large enough and a decay governed by the nonlinear term when p is greater than one but not so large.

Keywords: nonlocal diffusion, semilinear problems, asymptotic behaviour

Citation: Asymptotic Analysis, vol. 52, no. 1-2, pp. 143-155, 2007

Authors: Alexandrov, Oleg

Article Type: Research Article

Abstract: In this paper we find the far-field expansion of the outgoing Green's function in a three-dimensional cylindrical optical waveguide. As model for the field propagation we use the Helmholtz equation. We compare our results with the related paper (J. Inst. Math. Appl. 21(3) (1978), 315–330).

Keywords: wave propagation, optical waveguide, Green's function, asymptotic expansion

Citation: Asymptotic Analysis, vol. 52, no. 1-2, pp. 157-171, 2007