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 ε1/εi, their linear dependence over the integers and also, unexpectedly, by the rate in which the ratios ε1/εi 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.
|פורסם - מאי 1999
|פורסם באופן חיצוני