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# Communications in Mathematical Sciences

## Volume 20 (2022)

### Number 3

### On homogenization of the Landau–Lifshitz equation with rapidly oscillating material coefficient

Pages: 653 – 694

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n3.a3

#### Authors

#### Abstract

In this paper, we consider homogenization of the Landau–Lifshitz equation with a highly oscillatory material coefficient with period $\varepsilon$ modeling a ferromagnetic composite. We derive equations for the homogenized solution to the problem and the corresponding correctors and obtain estimates for the difference between the exact and homogenized solution as well as corrected approximations to the solution. Convergence rates in $\varepsilon$ over times $O(\varepsilon^\sigma)$ with $0 \leq \sigma \leq 2$ are given in the Sobolev norm $H^q$, where $q$ is limited by the regularity of the solution to the detailed Landau–Lifshitz equation and the homogenized equation. The rates depend on $q$, $\sigma$ and the number of correctors.

#### Keywords

homogenization, micromagnetics, magnetization dynamics, multiscale

#### 2010 Mathematics Subject Classification

35B27, 65M15, 82D40

Received 22 December 2020

Received revised 11 August 2021

Accepted 24 August 2021

Published 21 March 2022