Linearized viscoelastic Oldroyd fluid motion in an almost periodic environment

Abstract

In most of the linear homogenization problems involving convolution terms so far studied, the main tool used to derive the homogenized problem is the Laplace transform. Here we propose a direct approach enabling one to tackle both linear and nonlinear homogenization problems that involve convolution sequences without using Laplace transform. To illustrate this, we investigate in this paper the asymptotic behavior of the solutions of a Stokes-Volterra problem with rapidly oscillating coefficients describing the viscoelastic fluid flow in a fixed domain. Under the almost periodicity assumption on the coefficients of the problem, we prove that the sequence of solutions of our "-problem converges in L 2 to a solution of a rather classical Stokes system. One important fact is that the memory disappears in the limit. To achieve our goal, we use some very recent results about the sigma-convergence of convolution sequences.

Multiscale nonlocal flow in a fractured porous medium

“…The proof is very similar to the one of its homologue Theorem 2.6 in [44] (see also [43]). Since Theorem 2.6 in [44] involves almost periodicity and moreover is checked in the two-scale sense, it is suitable to repeat the proof here in the periodicity and multiscale frameworks for completeness. First and foremost, it is easy to see that the sequence (u ε * v ε ) ε∈E is bounded in L m (Ω).…”

“…[29,45]), our approach can handle more complicated homogenization problems with nonlocal terms in both time and space variables; see e.g. [44]. As far as we know, this is the first time that such a problem is considered in the literature.…”

Asymptotic behaviour of viscoelastic composites with almost periodic microstructures

“…A more general version of that result in the framework of Σ‐convergence has been for the first time stated and proved in , but with the restricted assumption that the open set is bounded. Still in the Σ‐convergence framework, a more general proof (in any open set ) has been given in the almost periodic setting in a very recent work . The use of this method instead of the Laplace transform method is motivated by at least two facts: (i) it enables one to tackle both linear and nonlinear homogenization problems involving convolution both in fast space and time variables , which is not the case for the latter method; (ii) as observed in numerical experiments by the authors of (see Section 5 therein), ‘the memory effects make numerical inverse Laplace transform very complicated’.…”

Nonlocal Stokes-Vlasov system: Existence and deterministic homogenization results

“…when f ε ≡ 0) the asymptotic analysis of Eqs. (1.1)- (1.5) reduces to the study of the asymptotics of (1.2)-(1.5) (with of course f ε ≡ 0 therein), which has been very recently undertaken by Woukeng [35] in the almost periodic framework.…”