Analysis of a three-dimensional phase-field model for solidification under a magnetic field effect

This paper presents a mathematical analysis to a three-dimensional isothermal model of solidification for a binary alloy with melt convection and under a magnetic field effect. The model consists of a highly non-linear system of partial differential equations for the state variables: the velocity field, the pressure, the potential function of the electrical field, the phase-field which represents the solid/liquid phase of the alloy, and the concentration. The well-posedness of the model is discussed. Moreover, the existence of solutions when the diffusion coefficient of the concentration equation vanishes for some values of the phase-field is investigated4821CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO - CNPQCOORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIOR - CAPES308093/2018-6; 402388/2016-0001This work was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. A.P. Ferreira was partially supported by Centro Federal de Educação Tecnológica de Minas Gerais. G. Planas was partially supported by CNPq-Brazil, grants 308093/2018-6 and 402388/2016-