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Scale-Transformations in the Homogenization of Nonlinear Magnetic Processes

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Abstract

A class of nonlinear first-order processes is formulated as a minimization principle. In the presence of oscillating data, a two-scale model is then derived, via Nguetseng’s notion of two-scale convergence. The dependence on the fine-scale variable is eliminated by averaging with respect to the fine-scale (scale-integration or upscaling); conversely, any two-scale solution is retrieved from a coarse-scale one (scale-disintegration or downscaling). These results are first developed in a general functional framework, and are then applied to the homogenization of a relaxation dynamics in magnetic composites:

$$ \mathcal{A}(x/\varepsilon) {\partial B_\varepsilon\over \partial t} + \alpha(B_\varepsilon, x/\varepsilon) \ni H_\varepsilon $$
$$ \nabla \cdot B_\varepsilon=0, \quad \nabla \times H_\varepsilon =J(x) \quad \forall\;\varepsilon > 0. $$

Here J is a prescribed current density. \({\mathcal{A}(y)}\) is a positive-definite symmetric tensor, α(·, y) is a maximal monotone operator; both are assumed to depend periodically on the fine-scale variable y. The homogenized problem consists in coupling the magnetostatic equations with a maximal monotone relation \({B\in \gamma(H)}\) that is local in space but not in time. This scale-transformation procedure is also interpreted in terms of (single-scale) Γ-convergence.

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  1. Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, 38050, Povo, Trento, Italy

    Augusto Visintin

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Visintin, A. Scale-Transformations in the Homogenization of Nonlinear Magnetic Processes. Arch Rational Mech Anal 198, 569–611 (2010). https://doi.org/10.1007/s00205-010-0296-8

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