Weak Solutions and Simulations for a Generalized Phase‐Field Crystal Model With Neumann Boundary Conditions

ABSTRACT We study a generalized phase‐field crystal (GPFC) model, which is a quasilinear parabolic equation of sixth‐order for an order parameter. The model is used to simulate the microstructure evolution in crystal growth, specifically focusing on the competition between square, hexagonal, and roll forms. Here, the global existence and uniqueness of weak solutions in three space dimensions are proved under Neumann boundary conditions by employing the Galerkin method. The rigorous connection between weak solutions to the PFC and the GPFC equations is established through an analysis of the asymptotic limit. Moreover, we carry out numerical simulations to validate the model.