Mathematical analysis of a model describing solute trapping during solidification of binary alloys

Abstract

In this paper, we present a mathematical analysis to a Cahn-Hilliard/Allen-Cahn system with degenerate mobility that models an isothermal process of solidification of a binary alloy. This model is able to predict an observable phenomenon called solute trapping. The existence of global weak solutions for the system is proved. We approximate the degenerate system and show the convergence of solutions to the approximated non-degenerate problem to a solution of the degenerate one. We also investigated deeply the non-degenerate system by showing the existence of global weak solutions, the existence of global strong solutions in the two-dimensional case, and local strong solutions in the three-dimensional case, as well as, providing conditions to the uniqueness be satisfied.

Keywords:

• Cahn-Hilliard/Allen-Cahn system
• degenerate mobility
• solute trapping

2010 Mathematics Subject Classifications:

• 35K52
• 35A01
• 80A22
• 82B26

Acknowledgments

This work was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. AFP was partially supported by Centro Federal de Educação Tecnológica de Minas Gerais. GP was partially supported by CNPq-Brazil, grants 308093/2018-6 and 402388/2016-0, and FAPESP-Brazil grant 19/02512-5.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. AFP was partially supported by Centro Federal de Educação Tecnológica de Minas Gerais. GP was partially supported by CNPq-Brazil, Conselho Nacional de Desenvolvimento Cientfico e Tecnolgico grants 308093/2018-6 and 402388/2016-0, and FAPESP-Brazil grant 19/02512-5.