Homogenization of the Heat Equation with a Vanishing Volumetric Heat Capacity
This paper is a study of 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. In particular, the volumetric heat capacity equals εq and the thermal conductivity oscillates with period ε in space and εr in time, where 0 < q < r are real numbers. By using certain evolution settings of multiscale and very weak multiscale convergence we investigate, as ε tends to zero, how the relation between the volumetric heat capacity and the microscopic structure affects the homogenized problem and its associated local problem. It turns out that this relation gives rise to certain special effects in the homogenization result.
- 1.Danielsson, T., Johnsen, P.: Homogenization of the heat equation with a vanishing volumetric heat capacity (2018). arXiv: 1809.11019Google Scholar
- 2.Flodén, L., Holmbom, A., Olsson Lindberg, M., Persson, J.: Homogenization of parabolic equations with an arbitrary number of scales in both space and time. J. Appl. Math. 2014, 16 pp. (2014)Google Scholar