Homogenization with multiple scales

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/ε_n). The essential question in this context is what is the weak limit of such functions when ε_i ↓ 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, α_i = lim ε₁/ε_i, their linear dependence over the integers and also, unexpectedly, by the rate in which the ratios ε₁/ε_i tend to their limit α_i. These results and the accompanying convergence rate estimates are then used in deriving the homogenized equations in each of the abovementioned problems.