Homogenization with multiple scales

Article type: Research Article

Authors: Tassa, Tamir^;

Affiliations: Department of Mathematics, University of California, Los Angeles, CA 90095‐1555, USA

Note: [] Present address: Algorithmic Research Ltd., 10 Nevatim st., Petach Tikva, Israel 49561. E‐mail: tamir@arx.com.

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(x,{x}/{{\varepsilon}_1},\ldots,{x}/{{\varepsilon}_n}). The essential question in this context is what is the weak limit of such functions when {\varepsilon}_i \downarrow 0 and what is the corresponding convergence rate. It is shown that the weak limit equals the average of f(x,\cdot) over an affine submanifold of the torus T^n; the submanifold and its dimension are determined by the limit ratios between the scales, \alpha_i= \lim {{\varepsilon}_1}/{{\varepsilon}_i}, their linear dependence over the integers and also, unexpectedly, by the rate in which the ratios {{\varepsilon}_1}/{{\varepsilon}_i} tend to their limit \alpha_i. These results and the accompanying convergence rate estimates are then used in deriving the homogenized equations in each of the abovementioned problems.

Journal: Asymptotic Analysis, vol. 20, no. 1, pp. 61-96, 1999