# Homogenized phase transition model for perforated ferromagnetic media

## TL;DR

This paper develops a rigorous homogenization approach for a phase-field model describing the paramagnetic-ferromagnetic transition in perforated media, leading to explicit formulas for the effective Curie temperature tensor.

## Contribution

It introduces a novel homogenization method for a complex nonlinear coupled system modeling phase transitions in perforated ferromagnetic materials.

## Key findings

- Derived effective equations for the phase transition in perforated media.
- Explicit calculation of the macroscopic Curie temperature tensor.
- Established a homogenization framework for doubly nonlinear coupled systems.

## Abstract

This work presents a rigorous prediction of the effective equations governing the paramagnetic-ferromagnetic phase transition in a perforated three-dimensional body. Assuming a periodic distribution of perforations, we investigate the asymptotic behavior of solutions to the equations describing the thermodynamic and electromagnetic properties of the material as the period of the microstructure tends to zero. The microscopic model is a phase-field model within the Ginzburg-Landau framework for second-order phase transitions, where the phase-field is directly related to the magnetization vector. This model couples a nonlinear equation for the magnetization with the quasi-static Maxwell system and another nonlinear equation for the temperature. The primary mathematical challenge lies in homogenizing these equations, which exhibit a complex doubly non-linear structure. Additionally, the extension operators used within the homogenization framework precludes the application of standard Aubin--Lions compactness arguments. Our analysis employs two-scale convergence in conjunction with a two-scale decomposition based on an appropriate dilation operator. The nonlinearities are primarily addressed by means of a variant of compensated compactness and a Vitali compactness argument. From the perspective of practical applications, this work enables the explicit calculation of a Curie temperature tensor, capturing at the macroscopic scale the coupled effect of the material's geometric structure and its magnetic permeability tensor.