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

Abstract

The main contribution of this paper is the homogenization of the linear parabolic equation ${\partial }_t{u}^{\epsilon }(x,t)-\nabla ·(a(x/{\epsilon }^{q_{\mathrm{1}}},...,x/{\epsilon }^{q_n},t/{\epsilon }^{r_{\mathrm{1}}},...,t/{\epsilon }^{r_m})\nabla u^{\epsilon }(x,t))=f(x,t)$ 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 $q_{\mathrm{1}}=\mathrm{1}$, $q_{\mathrm{2}}=\mathrm{2}$, and .