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The journal Asymptotic Analysis fulfills a twofold function. It aims at publishing original mathematical results in the asymptotic theory of problems affected by the presence of small or large parameters on the one hand, and at giving specific indications of their possible applications to different fields of natural sciences on the other hand.
Asymptotic Analysis thus provides mathematicians with a concentrated source of newly acquired information which they may need in the analysis of asymptotic problems.
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
Citation: Asymptotic Analysis, vol. 71, no. 3, pp. 123-162, 2011
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
Citation: Asymptotic Analysis, vol. 71, no. 3, pp. 163-183, 2011
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