Weak solutions and simulations to a square phase‐field crystal model with Neumann boundary conditions

Abstract

We consider an initial‐boundary value problem of a square phase‐field crystal (SPFC) model, which is a nonlinear parabolic equation of sixth‐order. The model is a variant of the so‐called phase‐field crystal (PFC) model, introduced by K. R. Elder et al. This variant is more suitable to model the square symmetry crystal dynamics on atomic length and diffusive time scales. Here we analyze the SPFC equation endowed with Neumann boundary conditions in a bounded space. We first prove the existence and uniqueness of weak solutions global in time to the initial‐boundary value problem in the three‐dimensional case. The proof is based on the Galerkin method. Then we prove that the problem admits a bounded absorbing set. A few numerical experiments of the model are also performed to simulate the evolution of crystal growth.