1. Steady states. There are many two component systems in which phase separation can be induced by rapidly cooling the system. Thus, if a two component system, which is spatially uniform at temperature T1, is rapidly cooled to a second sufficiently lower temperature T2, then the cooled system will separate into regions of higher and lower concentration. A phenomenological description of the beh...

Abstract In this article, we present a second-order in time implicit-explicit (IMEX) local discontinuous Galerkin (LDG) method for computing the Cahn-Hilliard equation, which describes the phase separation phenomenon. It is well-known that the Cahn-Hilliard equation has a nonlinear stability property, i.e., the free-energy functional decreases with respect to time. The discretized Cahn-Hilliard...

We extend the invariant manifold method for analyzing the asymptotics of dissipative partial differential equations on unbounded spatial domains to treat equations in which the linear part has order greater than two. One important example of this type of equation which we analyze in some detail is the Cahn-Hilliard equation. We analyze the marginally stable solutions of this equation in some de...

We study two novel decoupled energy-law preserving numerical schemes for solving the CahnHilliard-Darcy (CHD) system which models two-phase flow in porous medium or in a Hele-Shaw cell. In the first scheme, the velocity in the Cahn-Hilliard equation is treated explicitly so that the Darcy equation is completely decoupled from the Cahn-Hilliard equation. In the second scheme, an intermediate vel...

The bidomain model is the standard model describing electrical activity of the heart. Here we study the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain AllenCahn equation) in two spatial dimensions. In the bidomain Allen-Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the ...

We consider the Cahn-Hilliard equation in one space dimension with scaling parameter ε, i.e. ut = (W ′(u) − εuxx)xx, where W is a nonconvex potential. In the limit ε ↓ 0, under the assumption that the initial data are energetically well-prepared, we show the convergence to a Stefan problem. The proof is based on variational methods and exploits the gradient flow structure of the Cahn-Hilliard e...

In this paper, inspired from the study on denoising, segmentation and reconstruction in image processing, and combining with the theories of two phase flows, we introduce one class of initial-boundary value problem of the Cahn-Hilliard equation with nonlocal terms. Then, by using the Schauder fixed point theorem, we obtain the existence of weak solutions to this initial boundary value problem f...

We present a new numerical scheme for solving a conservative Allen–Cahn equation with a space–time dependent Lagrange multiplier. Since the well-known classical Allen–Cahn equation does not have mass conservation property, Rubinstein and Sternberg introduced a nonlocal Allen–Cahn equation with a time dependent Lagrange multiplier to enforce conservation of mass. However, with their model it is ...

We present a hybrid level–set–Cahn–Hilliard model for the simulation of two phase flow in micro structures. Our method combines accurate contact line dynamics in the Cahn–Hilliard equation with the computationally cheaper level set method in regions where not the full Cahn–Hilliard dynamics are necessary to describe the flow. This is realized by employing the Cahn–Hilliard model close to rigid ...