## 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/ε_{1}, . . . , 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 T^{n}; the submanifold and its dimension are determined by the limit ratios between the scales, α_{i} = lim ε_{1}/ε_{i}, their linear dependence over the integers and also, unexpectedly, by the rate in which the ratios ε_{1}/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.

Original language | English |
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Pages (from-to) | 61-96 |

Number of pages | 36 |

Journal | Asymptotic Analysis |

Volume | 20 |

Issue number | 1 |

State | Published - May 1999 |

Externally published | Yes |