We establish existence and non-existence results for entire solutions to the fractional Allen–Cahn equation in R, which vanish on helicoids and are invariant under screw motion. In addition, we prove that helicoids are surfaces with vanishing non-local mean curvature.

Using an approximate analytical solution of the Cahn-Hilliard equation describing the coalescence during a first order phase transition, we compute the characteristic time for one step of period doubling in Langer's self similar scenario for Ostwald ripening. As an application, we compute the thermodynamically stable period of a 1D modulated phase pattern.

Abstract: In a numerical study, we consider the Allen–Cahn equation with a double-obstacle constraint. The constraint is a multivalued function that is provided by the subdifferential of the indicator function on a closed interval. Therefore, performing a numerical simulation of our problem poses difficulties. We propose a new approximate method for the constraint and demonstrate its validity. ...

Eutectic freeze crystallisation is a promising way of purifying water for it may require less energy than other methods. In order to simulate such process, phase field models as Allen-Cahn and Cahn-Hilliard can be used. this paper, port-Hamiltonian formulation the equations used coupled heat conduction, which allows thermodynamically consistent system written with help entropy functional. secon...

In this second part of the series, we focus on approximating the Hele-Shaw problem via the Cahn-Hilliard equation ut + ∆(ε∆u − εf(u)) = 0 as ε ↘ 0. The primary goal of this paper is to establish the convergence of the solution of the fully discrete mixed finite element scheme proposed in [21] to the solution of the Hele-Shaw (Mullins-Sekerka) problem, provided that the HeleShaw (Mullins-Sekerka...

This paper studies spinodal decomposition in the Cahn-Hilliard model on the unit disk. It has previously been shown that starting at initial conditions near a homogeneous equilibrium on a rectangular domain, solutions to the linearized and the nonlinear Cahn-Hilliard equation behave indistinguishably up to large distances from the homogeneous state. In this paper we demonstrate how these result...

Using the advective Cahn–Hilliard equation as a model, we illuminate the role of advection in phase-separating binary liquids. The advecting velocity is either prescribed, or is determined by an evolution equation that accounts for the feedback of concentration gradients into the flow. After obtaining some general results about the existence and regularity of solutions to the model equation, we...

Results are presented from numerical experiments aiming at the computation of stochastic phase-field models for phase transformations by coarse-graining molecular dynamics. The studied phase transformations occur between a solid crystal and a liquid. Nucleation and growth, sometimes dendritic, of crystal grains in a sub-cooled liquid is determined by diffusion and convection of heat, on the mac...