Heat flow in a random medium and homogenization.
Ian Frederick Pulizzotto · Deep Blue (University of Michigan) · 2001
Throughout this paper, we study heat flow in random versus homogeneous media. The main goal is to study in what sense the temperature uepsilon(x, o) in a medium with random d by d conductivity matrix a converges to the temperature u(x) of a medium with a constant (effective) d by d conductivity matrix q (where o represents randomness and x represents position in d-dimensional space). Assume a has uniformly elliptic symmetric part; this guarantees that heat always flows from warmer to cooler regions and never vice versa. In the one dimensional case (d = 1), explicit solutions of random and deterministic ODE's are derived. From these explicit solutions and two ergodic theorems, some convergence and nonconvergence results follow. It turns out that q11, is not the expectation of a11, but rather 1 E1/a11 . In the case d ≥ 3, variational solutions are constructed for the constant conductivity PDE, the random conductivity PDE, and the effective conductivity matrix q. Then using Green's functions, the Von Neumann ergodic theorem, and the construction of the Fourier transforms of uepsilon and u (in x), we prove a uniform convergence and a fractional-derivative-convergence result. In doing so, we extend Koslov's convergence-in-variance result. The nonconvergence result in the one dimensional case and the correction term in the Papanicolaou-Varadhan strong convergence theorem both suggest that our fractional-derivative-convergence result is tight.