Geodesic
Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales
Mathematica Bohemica, Tome 146 (2021) no. 4, pp. 483-511
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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
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.
DOI : 10.21136/MB.2021.0087-19
Classification : 35B27, 35K20
Keywords: homogenization; parabolic problem; multiscale convergence; very weak multiscale convergence; two-scale convergence 
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Danielsson, Tatiana; Johnsen, Pernilla. Homogenization of linear parabolic equations with three spatial and three temporal scales for certain matchings between the microscopic scales. Mathematica Bohemica, Tome 146 (2021) no. 4, pp. 483-511. doi: 10.21136/MB.2021.0087-19

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