Homogenization with multiple scales

Tamir Tassa

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

We study the homogenization of oscillatory solutions of partial differential equations with a multiple number of small scales. We consider a variety of problems - nonlinear convection-diffusion equations with oscillatory initial and forcing data, the Carleman model for the discrete Boltzman equations, and two-dimensional linear transport equations with oscillatory coefficients. In these problems, the initial values, force terms or coefficients are oscillatory functions with a multiple number of small scales - f(cursive Greek chi, cursive Greek chi/ε1, . . . , cursive Greek chi/εn). The essential question in this context is what is the weak limit of such functions when εi ↓ 0 and what is the corresponding convergence rate. It is shown that the weak limit equals the average of f(cursive Greek chi, ·) over an affine submanifold of the torus Tn; the submanifold and its dimension are determined by the limit ratios between the scales, αi = lim ε1i, their linear dependence over the integers and also, unexpectedly, by the rate in which the ratios ε1i tend to their limit αi. These results and the accompanying convergence rate estimates are then used in deriving the homogenized equations in each of the abovementioned problems.

Original languageEnglish
Pages (from-to)61-96
Number of pages36
JournalAsymptotic Analysis
Volume20
Issue number1
StatePublished - 1 May 1999
Externally publishedYes

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