Heat flow in a random medium and homogenization.

Abstract

Throughout this paper, we study heat flow in random versus homogeneous media. The main goal is to study in what sense the temperature <italic> u</italic>_epsilon(<italic>x</italic>, o) in a medium with random <italic>d</italic> by <italic>d</italic> conductivity matrix <bold> a</bold> converges to the temperature <italic>u</italic>(<italic>x</italic>) of a medium with a constant (effective) <italic>d</italic> by <italic>d</italic> conductivity matrix <bold>q</bold> (where o represents randomness and <italic> x</italic> represents position in <italic>d</italic>-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 (<italic>d</italic> = 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 <italic>q</italic>₁₁, is not the expectation of <italic>a</italic>₁₁, but rather 1<de>E1/a11</fen> </de> . In the case <italic>d</italic> &ge; 3, variational solutions are constructed for the constant conductivity PDE, the random conductivity PDE, and the effective conductivity matrix <bold>q</bold>. Then using Green's functions, the Von Neumann ergodic theorem, and the construction of the Fourier transforms of <italic> u</italic>_epsilon and <italic>u</italic> (in <italic>x</italic>), 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.