Solutions to a coupled Cahn–Hilliard-phase-field-crystal model for grain boundary motion

Abstract

In this paper, we consider an initial–boundary value problem (IBVP) of a coupled Cahn–Hilliard-phase-field-crystal (CH-PFC) system subject to homogeneous Neumann boundary conditions in one spatial dimension. This CH-PFC model couples the composition field of a diffusing species with the crystallographic and can be used to model the diffusion-induced grain boundary migration in crystalline materials. Under suitable assumptions on the coefficients and initial data, we prove that the IBVP possesses a global weak solution. Our existence proof, which contributes to the verification of the model, is only valid in one space dimension.

Keywords:

• Weak solutions
• degenerate mobility
• higher order parabolic equations
• Galerkin method

Mathematics Subject Classifications:

• 35D30
• 35K61

Acknowledgments

The authors wish to thank the anonymous reviewers for their careful reading and providing invaluable suggestions. The authors would like to sincerely appreciate Prof. Peicheng Zhu for his careful instruction.

Disclosure statement

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

Additional information

Funding

This work was partly supported by Science and Technology Commission of Shanghai Municipality (Grant No. 20JC1413600).