The Well - Posedness And Asymptotic Behavior Of The Solution Of The Three - Dimensional Phase - Field Crystal Model

Abstract:The phase-field crystal (PFC) model was recently proposed as a new and important approach to simulating crystals at the atomic scale in space but on a coarse-grained diffusive time scale. A variety of applications for the PFC approach can be seen in Physics. We consider in this paper the three dimension phase-field crystal equation with Neumann boundary conditions. We prove the existence of global attractors in Before that, we prove the unique existence of global weak solutions to the three dimension PFC equation based on the Galerkin approaching method. By proving the existence of a compact absorbing set, we shall show the problem admits a global attractor in Hr3(Ω). In the last chapter, we shall prove the existence of an exponential attractor, which has an explicit (exponential) control on the rate of attraction of the trajectories. The finite fractal dimension of the exponential attractor also derives the finite-dimensionality of the global attractor...