Homogenization of monotone parabolic problems with several temporal scales

Abstract

In this paper we homogenise monotone parabolic problems with two spatial scales and finitely many temporal scales. Under a certain well-separatedness assumption on the spatial and temporal scales as explained in the paper, we show that there is an H-limit defined by at most four distinct sets of local problems corresponding to slow temporal oscillations, slow resonant spatial and temporal oscillations (the "slow" self-similar case), rapid temporal oscillations, and rapid resonant spatial and temporal oscillations (the "rapid" self-similar case), respectively.

Homogenization of the Heat Equation with a Vanishing Volumetric Heat Capacity

“…Theorem 5 and the fact that u 1 is independent of s gives ΩT Y1,1 a (y, s) (∇u (x, t) + ∇ y u 1 (x, t, y)) v 1 (x) · ∇ y v 2 (y) c 1 (t) dydsdxdt = 0 and by the Variational lemma we have Y S a (y, s) ds (∇u (x, t) + ∇ y u 1 (x, t, y)) · ∇ y v 2 (y) dy = 0 a.e. in Ω T , which is the weak form of (18).…”

“…The definition was first given by Persson, see e.g. [18] where a more technically formulated version is given.…”

“…For r > q + 2, u 1 is determined by the elliptic local problem − ∇ y · S a (y, s) ds · (∇u (x, t) + ∇ y u 1 (x, t, y)) = 0 (18) and since u 1 is independent of s, the coefficient (15) can, in this case, be expressed as b∇u (x, t) = Y S a (y, s) ds (∇u (x, t) + ∇ y u 1 (x, t, y)) dy.…”

Homogenization of Linear Parabolic Equations with a Certain Resonant Matching Between Rapid Spatial and Temporal Oscillations in Periodically Perforated Domains

“…For the evolution setting we need the equivalent for multiscale convergence with time-dependent effect. Following [37] we provide the concept in the next definition.…”

“…In 2009, these results were extended by Woukeng, who studied non-linear parabolic problems with the same choice of scales in [41]. Also [37], by Persson, deals with monotone parabolic problems, but with an arbitrary number of temporal microscales, where none of them has to be identical with the rapid spatial scale or even has to be a power of ε. In [21] we return to the case of linear parabolic homogenization for arbitrary numbers of spatial and temporal scales benefitting from the concept of jointly separated scales introduced in [35].…”