Asymptotic Analysis - Volume 87, issue 1-2

Authors: Chechkin, G.A. | D'Apice, C. | De Maio, U. | Piatnitski, A.L.

Article Type: Research Article

Abstract: In the paper we deal with the homogenization problem for the Poisson equation in a singularly perturbed domain with multilevel oscillating boundary. This domain consists of the body, a large number of thin periodically situated cylinders joining to the body through thin random transmission zone with rapidly oscillating boundary. Inhomogeneous Fourier boundary conditions with perturbed coefficients are set on the boundaries of the thin cylinders and on the boundary of the transmission zone. We prove the homogenization theorems. Moreover we derive estimates of deviation of the solution to initial problem from the solution to the homogenized problem in different cases. …It appears that depending on small parameters in Fourier boundary conditions of initial problem one can obtain Dirichlet, Neumann or Fourier boundary conditions in the homogenized problem. We estimate the convergence of solutions in these three cases. Show more

Keywords: homogenization, estimates of convergence, rapidly oscillating boundary, singular perturbations, random structures

DOI: 10.3233/ASY-131194

Citation: Asymptotic Analysis, vol. 87, no. 1-2, pp. 1-28, 2014

Authors: Bonetti, Elena | Frémond, Michel

Article Type: Research Article

Abstract: We build a predictive theory for the evolution of mixture of helium and supercooled helium at low temperature. The absolute temperature θ and the volume fraction β of helium, which is dominant at temperature larger than the phase change temperature, are the state quantities. The predictive theory accounts for local interactions at the microscopic level, involving the gradient of β. The nonlinear heat flux in the supercooled phase results from a Norton–Hoff potential. We prove that the resulting set of partial differential equations has solutions within a convenient analytical frame.

Keywords: supercooled helium, phase change, predictive theory, existence theorem

DOI: 10.3233/ASY-131195

Citation: Asymptotic Analysis, vol. 87, no. 1-2, pp. 29-42, 2014

Authors: Hu, Wenqing | Tcheuko, Lucas

Article Type: Research Article

Abstract: We study the asymptotic behavior of a diffusion process with small diffusion in a domain D. This process is reflected at ∂D with respect to a co-normal direction pointing inside D. Our asymptotic result is used to study the long time behavior of the solution of the corresponding parabolic PDE with Neumann boundary condition.

Keywords: PDE with a small parameter, large deviations, Freidlin–Wentzell theory, diffusion process with reflection

DOI: 10.3233/ASY-131197

Citation: Asymptotic Analysis, vol. 87, no. 1-2, pp. 43-56, 2014

Authors: Hajaiej, H.

Article Type: Research Article

Abstract: We prove the orbital stability and then characterize the orbit of standing waves of Schrödinger equation of Hartree type.

Keywords: orbit, standing waves, NLS, characterization

DOI: 10.3233/ASY-131199

Citation: Asymptotic Analysis, vol. 87, no. 1-2, pp. 57-64, 2014

Authors: Almada, Sergio Angel | Spiliopoulos, Konstantinos

Article Type: Research Article

Abstract: In this paper we study the fluctuations from the limiting behavior of small noise random perturbations of diffusions with multiple scales. The result is then applied to the exit problem for multiscale diffusions, deriving the limiting law of the joint distribution of the exit time and exit location. We apply our results to the first order Langevin equation in a rough potential, studying both fluctuations around the typical behavior and the conditional limiting exit law, conditional on the rare event of going against the underlying deterministic flow.

Keywords: multiscale dynamics, small noise, Gaussian correction, exit problem, conditional rare events

DOI: 10.3233/ASY-131207

Citation: Asymptotic Analysis, vol. 87, no. 1-2, pp. 65-90, 2014

Authors: Griso, G.

Article Type: Research Article

Abstract: In this paper we investigate the homogenization problem with a non-homogeneous Dirichlet condition. Our aim is to give error estimates with boundary data in H1/2 (∂Ω). The tools used are those of the unfolding method in periodic homogenization.

Keywords: periodic homogenization, error estimate, non-homogeneous Dirichlet condition, periodic unfolding method

DOI: 10.3233/ASY-131200

Citation: Asymptotic Analysis, vol. 87, no. 1-2, pp. 91-121, 2014