George Papanicolaou

This is a reprinting of a book originally published in 1978. At that time it was the first book on the subject of homogenization, which is the asymptotic analysis of partial differential equations with rapidly oscillating coefficients, and as such it sets the stage for what problems to consider and what methods to use, including probabilistic methods. At the time the book was written the use of asymptotic expansions with multiple scales was new, especially their use as a theoretical tool, combined with energy methods and the construction of test functions for analysis with weak convergence met

We derive and analyze transport equations for the energy density of waves of any kind in a random medium. The equations take account of nonuniformities of the background medium, scattering by random inhomogeneities, polarization effects, coupling of different types of waves, etc. We also show that diffusive behavior occurs on long time and distance scales and we determine the diffusion coefficients. The results are specialized to acoustic, electromagnetic, and elastic waves. The analysis is based on the governing equations of motion and uses the Wigner distribution.

A perturbation analysis with respect to the space dimension is used to construct singular solutions of the two-dimensional Schr\"odinger equation with cubic nonlinearity. These solutions blow up at a rate {ln ln[(${\mathit{t}}^{\mathrm{*}}$-t${)}^{\mathrm{\ensuremath{-}}1}$]/(${\mathit{t}}^{\mathrm{*}}$-t)${\mathrm{}}}^{1/2}$, in contrast to the behavior in three dimensions where there is no logarithmic correction. The form of such solutions is supported by the results of high-resolution numerical simulations.