We consider the spectrum of the linear operator that arises upon linearization of the Cahn–Hilliard equation in dimensions d ≥ 2 about a planar transition front (a solution that depends on only one distinguished space variable and that has different values at ±∞). In previous work the author has established conditions on this spectrum under which such planar transition fronts are asymptotically...

An efficient semi-implicit Fourier spectral method is implemented to solve the Cahn-Hilliard equation with a variable mobility. The method is orders of magnitude more efficient than the conventional forward Euler finite-difference method, thus allowing us to simulate large systems for longer times. We studied the coarsening kinetics of interconnected two-phase mixtures using a Cahn-Hilliard equ...

We present an unconditionally stable second-order hybrid numerical method for solving the Allen–Cahn equation representing amodel for antiphase domain coarsening in a binary mixture. The proposed method is based on operator splitting techniques. The Allen–Cahn equation was divided into a linear and a nonlinear equation. First, the linear equation was discretized using a Crank–Nicolson scheme an...

Validated continuation was introduced in [4] as means of checking that the classical continuation method applied to a Galerkin projection of a PDE provides a locally unique equilibrium to the PDE of interest. In this paper we extend the numerical technique to include a parameter that leads to better bounds on the errors associated with the Galerkin truncation. We test this method on the Swift-H...

The Cahn-Hilliard equation has its origin in material sciences and serves as a model for phase separation and phase coarsening in binary alloys. A new approach in the class of fourth order inpainting algorithms is inpainting of binary images using the Cahn-Hilliard equation. We will present a generalization of this fourth order approach for grayvalue images. This is realized by using subgradien...

In this article, we are interested in the study of the asymptotic behavior, in terms of finite-dimensional attractors, of a generalization of the Cahn-Hilliard equation with a fidelity term (integrated over Ω\D instead of the entire domain Ω, D ⊂⊂ Ω). Such a model has, in particular, applications in image inpainting. The difficulty here is that we no longer have the conservation of mass, i.e. o...

We study the asymptotic dynamics of the Cahn–Hilliard equation via the “Gamma-convergence” of gradient flows scheme initiated by Sandier and Serfaty. This gives rise to an H1-version of a conjecture by De Giorgi, namely, the slope of the Allen–Cahn functional with respect to the H−1-structure Gamma-converges to a homogeneous Sobolev norm of the scalar mean curvature of the limiting interface. W...