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

2019/08/16 by Tatiana Danielsson, Danielsson, Tatiana, Pernilla Johnsen +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1908.05892

openalex publication_date 2019/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we establish compactness results of multiscale and very weak\nmultiscale type for sequences bounded in L²(0,T;H₀¹(\Ω )),\nfulfilling a certain condition. We apply the results in the homogenization of\nthe parabolic partial differential equation \ε\n^p\∂_tu_\ε \( x,t\) -\∇ \⋅ \( a\(\nx/\ε ,x/\ε ²,t/\ε^q,t/\ε ^r\)\n\∇ u_\ε \( x,t\)\) = f\( x,t\) , where\n0<p<q<r. The homogenization result reveals two special phenomena, namely that\nthe homogenized problem is elliptic and that the matching for when the local\nproblem is parabolic is shifted by p, compared to the standard matching that\ngives rise to local parabolic problems.\n

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