Homogenization of the Heat Equation with a Vanishing Volumetric Heat Capacity

Abstract

This paper is devoted to the homogenization of the heat conduction equation, with a homogeneous Dirichlet boundary condition, having a periodically oscillating thermal conductivity and a vanishing volumetric heat capacity. A homogenization result is established by using the evolution settings of multiscale and very weak multiscale convergence. In particular, we investigate how the relation between the volumetric heat capacity and the microscopic structure effects the homogenized problem and its associated local problem. It turns out that the properties of the microscopic geometry of the problem give rise to certain special effects in the homogenization result.

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

“…For the homogenization part of this paper we apply the convergence results to establish a homogenization result for (1) with 13 different outcomes, depending on the choices of parameters p, q and r. The homogenization result will reveal two phenomena, which also occurred in both [13] and the proceeding work [6], where the homogenization of parabolic problems of a similar kind, but with only one rapid scale in space and time each, was presented. The first phenomenon is that the homogenized problem is of elliptic type even though the original problem is a parabolic one and the second is that resonance occurs for different matchings between the microscopic scales than the standard ones.…”