Very weak multiscale convergence

Homogenization of the Heat Equation with a Vanishing Volumetric Heat Capacity

“…for all v ∈ H 1 0 (Ω) and c ∈ D(0, T ). We will now show that the solution to (6) is bounded in L 2 (0, T ; H 1 0 (Ω)), i.e. it satisfies the a priori estimate u ε L 2 (0,T ;H 1 0 (Ω)) ≤ C,…”

“…In [15], further progress in the context of Σ-convergence led to a closely related result and a simplification for the applicability in the homogenization procedure. The present form of the concept was given for an arbitrary number of spatial scales in [6], where also the name "very weak multiscale convergence" was established. Following [17] and [9], we give the evolution version of very weak multiscale convergence including arbitrarily many spatial and temporal scales.…”

A Note on Parabolic Homogenization with a Mismatch between the Spatial Scales

“…To treat evolution problems with fast time oscillations, such as (1), we also need the concept of very weak multiscale convergence, see, for example, [2,5].…”

“…Hence, the problem is not of a reiterated type. We prove by means of very weak multiscale convergence [2] that the corrector 2 associated with the gradient for the second rapid spatial scale 2 actually vanishes. Already, in [3,4], it was observed that having more than one rapid temporal scale in parabolic problems does not generate a reiterated problem and in this paper we can see that nor does the addition of a spatial scale if it is contained in a coefficient that is multiplied with the time derivative of .…”

“…Parabolic homogenization problems for ≡ 1 have been studied for different combinations of spatial and temporal scales in several papers by means of techniques of two-scale convergence type with approaches related to the one first introduced in [5], see, for example, [2,3,[6][7][8], and in, for example, [9][10][11], techniques not of two-scale convergence type are applied. Concerning cases where, as in (1) above, we do not have ≡ 1, Nandakumaran and Rajesh [12] studied a nonlinear parabolic problem with the same frequency of oscillation in time and space, respectively, in the elliptic part of the equation and an operator oscillating in space with the same frequency appearing in the temporal differentiation term.…”