Homogenization of parabolic equations with an arbitrary number of scales in both space and time. (English) Zbl 1406.35140

Summary: The main contribution of this paper is the homogenization of the linear parabolic equation \(\partial_tu^{\varepsilon}(x,t)-\nabla\cdot(a(x/\varepsilon^{q_1},\ldots,x/\varepsilon^{q_n},t/\varepsilon^{r_1},\ldots,t/\varepsilon^{r_m})\nabla u^{\varepsilon}(x,t))=f(x,t)\) exhibiting an arbitrary finite number of both spatial and temporal scales. We briefly recall some fundamentals of multiscale convergence and provide a characterization of multiscale limits for gradients, in an evolution setting adapted to a quite general class of well-separated scales, which we name by jointly well-separated scales (see appendix for the proof). We proceed with a weaker version of this concept called very weak multiscale convergence. We prove a compactness result with respect to this latter type for jointly well-separated scales. This is a key result for performing the homogenization of parabolic problems combining rapid spatial and temporal oscillations such as the problem above. Applying this compactness result together with a characterization of multiscale limits of sequences of gradients we carry out the homogenization procedure, where we together with the homogenized problem obtain \(n\) local problems, that is, one for each spatial microscale. To illustrate the use of the obtained result, we apply it to a case with three spatial and three temporal scales with \(q_1=1\), \(q_2=2\), and \(0<r_1<r_2\).


35K10 Second-order parabolic equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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