Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales

Danielsson, Tatiana ; Johnsen, Pernilla

Mathematica Bohemica, Tome 146 (2021) no. 4, pp. 483-511.

Résumé

In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in $L^{2}(0,T;H_{0}^{1}(\Omega ))$, fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation $\varepsilon ^{p}\partial _{t}u_{\varepsilon }(x,t) -\nabla \cdot ( a( x\varepsilon ^{-1} ,x\varepsilon ^{-2},t\varepsilon ^{-q},t\varepsilon ^{-r}) \nabla u_{\varepsilon }(x,t) ) = f(x,t) $, where $0$. The homogenization result reveals two special phenomena, namely that the homogenized problem is elliptic and that the matching for which the local problem is parabolic is shifted by $p$, compared to the standard matching that gives rise to local parabolic problems.

Détail

DOI : 10.21136/MB.2021.0087-19

Classification : 35B27, 35K20
Keywords: homogenization; parabolic problem; multiscale convergence; very weak multiscale convergence; two-scale convergence