Detection of scales of heterogeneity ‎and parabolic homogenization applying ‎very weak multiscale convergence

Abstract

We apply a new version of multiscale convergence named very weak multiscale convergence to find possible frequencies of oscillation in an unknown coefficient of a partial differential equation from its solution. We also use this notion to study homogenization of a certain linear parabolic problem with multiple spatial and temporal scales.

Homogenization of nonlinear dissipative hyperbolic problems exhibiting arbitrarily many spatial and temporal scales

“…by (B6) and the Generalized Hölder Inequality which is applicable since it holds that 1 2(γ+1)/γ + 1 2(γ+1) + 1 2 = 1. Letting ε → 0 and using (8) in (9) we obtain…”

“…Since we study the problem (1) exhibiting an arbitrary number of spatial and temporal scales we invoke the concept of evolution multiscale convergence, a generalization of Nguetseng's classical twoscale convergence [18]. We give the following definition, also exploited in, e.g., [9] and [10]. Definition 2.1.…”

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

“…The method was further developed by Allaire [16] and generalized to multiple scales by Allaire and Briane [1]. To homogenize problem (1), we use the further generalization in the definition below, adapted to evolution settings, see, for example, [8].…”

“…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.…”

Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time

“…Significant progress was made in [11], where the case with an arbitrary number of temporal scales is treated and none of them has to coincide with the single fast spatial scale. A first study of parabolic problems where the number of fast spatial and temporal scales both exceeds one is found in [12], where the fast spatial scales are 1 = , 2 = 2 and the rapid temporal scales are chosen as 1 = 2 , 2 = 4 , and 3 = 5 . Similar techniques have also been recently applied to hyperbolic problems.…”