Asymptotic Analysis - Volume 90, issue 1-2

Authors: Boryc, Marcin | Komorowski, Tomasz

Article Type: Research Article

Abstract: In the present paper we prove homogenization for diffusions and the solutions of their corresponding Kolmogorov's equation with random, non-stationary, but locally ergodic coefficients. The results generalizes earlier work of Papanicolaou and Varadhan [in: Statistics and Probability: Essays in Honor of C.R. Rao, North-Holland, 1982, pp. 547–552] and Yurinskii [Sibirsk. Mat. Zh. 23(2) (1982), 176–188].

Keywords: homogenization, random diffusions, non-stationary environments, local ergodicity

DOI: 10.3233/ASY-141221

Citation: Asymptotic Analysis, vol. 90, no. 1-2, pp. 1-20, 2014

Authors: Tachim Medjo, T.

Article Type: Research Article

Abstract: This article studies the pullback asymptotic behavior of solutions for a non-autonomous Cahn–Hilliard–Navier–Stokes (CH–NS) system in a two-dimensional domain. We prove the existence of pullback attractors 𝒜VM in VM (the velocity has the H1 -regularity) and 𝒜YM in YM (the velocity has the L2 -regularity). Then we verify the regularity of the pullback attractors by proving that 𝒜VM =𝒜YM , which implies the pullback asymptotic smoothing effect of the model in the sense that the solutions eventually become more regular than the initial data.

Keywords: pullback attractor, non-autonomous two-phase flow, cocycle

DOI: 10.3233/ASY-141225

Citation: Asymptotic Analysis, vol. 90, no. 1-2, pp. 21-51, 2014

Authors: Bělík, Pavel | Dokken, Douglas P. | Scholz, Kurt | Shvartsman, Mikhail M.

Article Type: Research Article

Abstract: We consider a modification of the fluid flow model for a tornado-like swirling vortex developed by Serrin [Phil. Trans. Roy. Soc. London, Series A, Math & Phys. Sci. 271(1214) (1972), 325–360], where velocity decreases as the reciprocal of the distance from the vortex axis. Recent studies, based on radar data of selected severe weather events [Mon. Wea. Rev. 133(9) (2005), 2535–2551; Mon. Wea. Rev. 128(7) (2000), 2135–2164; Mon. Wea. Rev. 133(1) (2005), 97–119], indicate that the angular momentum in a tornado may not be constant with the radius, and thus suggest a different scaling of the velocity/radial distance dependence. …Motivated by this suggestion, we consider Serrin's approach with the assumption that the velocity decreases as the reciprocal of the distance from the vortex axis to the power b with a general b>0. This leads to a boundary-value problem for a system of nonlinear differential equations. We analyze this problem for particular cases, both with nonzero and zero viscosity, discuss the question of existence of solutions, and use numerical techniques to describe those solutions that we cannot obtain analytically. Show more

Keywords: Serrin's swirling vortex, Navier–Stokes equations, Euler equations, Cai's power law, tornado modeling

DOI: 10.3233/ASY-141228

Citation: Asymptotic Analysis, vol. 90, no. 1-2, pp. 53-82, 2014

Authors: Wang, Xiang-Sheng

Article Type: Research Article

Abstract: In this paper, we introduce an innovative and systematic technique to study delay differential equations via polynomials. First, we review an intrinsic relation between delay differential equations and polynomials. From this relation, we obtain long time behaviors of the solutions to delay differential equations via asymptotic analysis of the corresponding polynomials. Moreover, we derive asymptotic formulas and upper bounds for the intrinsic growth rate of delay differential equations, as well as a Gronwall-type inequality for delay differential inequalities.

Keywords: delay differential equations, asymptotic analysis, Gronwall-type inequality, intrinsic growth rate

DOI: 10.3233/ASY-141232

Citation: Asymptotic Analysis, vol. 90, no. 1-2, pp. 83-103, 2014

Authors: Jollivet, Alexandre

Article Type: Research Article

Abstract: We define scattering data for the Newton equation in a potential V∈C2 (Rn ,R), n≥2, that decays at infinity like r−α for some α∈(0,1]. We provide estimates on the scattering solutions and scattering data and we prove, in particular, that the scattering data at high energies uniquely determine the short range part of the potential up to the knowledge of the long range tail of the potential. The Born approximation at fixed energy of the scattering data is also considered. We then change the definition of the scattering data and consider also inverse scattering in other asymptotic regimes. These …results were obtained by developing the inverse scattering approach of Novikov [Ark. Mat. 37 (1999), 141–169]. Show more

Keywords: inverse scattering at high energies, Newton equation in a long range potential, inverse problems

DOI: 10.3233/ASY-141244

Citation: Asymptotic Analysis, vol. 90, no. 1-2, pp. 105-132, 2014

Authors: Obrecht, Caroline

Article Type: Research Article

Abstract: By a modulational analysis, we derive a Benney–Roskes type asymptotic model to the water wave equation with surface tension. In contrast to the classical Benney–Roskes system, this model preserves – at least for high frequencies – the dispersion relation of the full water wave equations. The formal derivation of the model is completed by some considerations on its consistency with the water wave equations.

Keywords: water wave asymptotics, Davey–Stewartson system, full dispersion, well-posedness

DOI: 10.3233/ASY-141246

Citation: Asymptotic Analysis, vol. 90, no. 1-2, pp. 133-160, 2014

Authors: Rahmani, Leila | Vial, Grégory

Article Type: Research Article

Abstract: In this paper, we consider a partially clamped plate which is stiffened on a portion of its free boundary. Our aim is to build an asymptotic expansion of the displacement, solution of the Kirchhoff–Love model, with respect to the thickness of the stiffener. Due to the mixed boundary conditions, singularities appear, obstructing the construction of the terms of the asymptotic expansion in the same way as if the plate was surrounded by the stiffener on its whole boundary. Using a splitting into regular and singular parts, we are able to formulate an asymptotic expansion involving profiles which allow to take …into account the singularities. Show more

Keywords: thin plate, stiffener, asymptotic expansion, elliptic partial differential equation, corner singularities

DOI: 10.3233/ASY-141236

Citation: Asymptotic Analysis, vol. 90, no. 1-2, pp. 161-187, 2014