Asymptotic Analysis - Volume 71, issue 3

Authors: Cavaterra, Cecilia | Gal, Ciprian G. | Grasselli, Maurizio

Article Type: Research Article

Abstract: We consider a modified Cahn–Hiliard equation where the velocity of the order parameter u depends on the past history of Δμ, μ being the chemical potential with an additional viscous term αut , α≥0. This type of equation has been proposed by P. Galenko et al. to model phase separation phenomena in special materials (e.g., glasses). In addition, the usual no-flux boundary condition for u is replaced by a nonlinear dynamic boundary condition which accounts for possible interactions with the boundary. The resulting boundary value problem is subject to suitable initial conditions and is reformulated in the so-called past history …space. Existence of a variational solution is obtained. Then, in the case α>0, we can also prove uniqueness and construct a strongly continuous semigroup acting on a suitable phase space. We show that the corresponding dynamical system has a (smooth) global attractor as well as an exponential attractor. In the case α=0, we only establish the existence of a trajectory attractor. Show more

Keywords: Cahn–Hilliard equations, dynamic boundary conditions, global attractors, exponential attractors, trajectory attractors, memory relaxation

DOI: 10.3233/ASY-2010-1019

Citation: Asymptotic Analysis, vol. 71, no. 3, pp. 123-162, 2011

Authors: Neukamm, Stefan | Stelzig, Philipp Emanuel

Article Type: Research Article

Abstract: We study the effects of translation on two-scale convergence. Given a two-scale convergent sequence (uε (x))ε with two-scale limit u(x,y), then in general the translated sequence (uε (x+t))ε is no longer two-scale convergent, even though it remains two-scale convergent along suitable subsequences. We prove that any two-scale cluster point of the translated sequence is a translation of the original limit and has the form u(x+t,y+r) where the microscopic translation r belongs to a set that is determined solely by t and the vanishing sequence (ε). Finally, we apply this result to a novel homogenization problem that involves two …different coordinate frames and yields a limiting behavior governed by emerging microscopic translations. Show more

Keywords: two-scale convergence, translation, homogenization, Γ-convergence

DOI: 10.3233/ASY-2010-1017

Citation: Asymptotic Analysis, vol. 71, no. 3, pp. 163-183, 2011