An error bound is proved for a fully practical piecewise linear nite element approximation, using a backward Euler time discretization, of the Cahn-Hilliard equation with a logarithmic free energy.

A boundary control problem for the viscous Cahn–Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved.

New types of stationary solutions of a one-dimensional driven sixthorder Cahn-Hilliard type equation that arises as a model for epitaxially growing nano-structures such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the hu...

This paper studies finite element approximations of the stochastic Allen–Cahn equation with gradient-type multiplicative noise that is white in time and correlated in space. The sharp interface limit—as the diffuse interface thickness vanishes—of the stochastic Allen–Cahn equation is formally a stochastic mean curvature flow which is described by a stochastically perturbed geometric law of the ...

We present an efficient time-stepping scheme for simulations of the coupled Navier– Stokes Cahn–Hilliard equations for the phase field approach. The scheme has several attractive characteristics: (i) it is suitable for large density ratios, and numerical experiments with density ratios up to 1000 have been presented; (ii) it involves only constant (time-independent) coefficient matrices for all...

We study two-dimensional (2D) binary-fluid turbulence by carrying out an extensive direct numerical simulation (DNS) of the forced, statistically steady turbulence in the coupled Cahn-Hilliard and Navier-Stokes equations. In the absence of any coupling, we choose parameters that lead (a) to spinodal decomposition and domain growth, which is characterized by the spatiotemporal evolution of the C...

The goal of this paper is to solve a long standing open problem, namely, the asymptotic development of order 2 by Γ-convergence of the mass-constrained Cahn–Hilliard functional. This is achieved by introducing a novel rearrangement technique, which works without Dirichlet boundary conditions.

We study the limiting behavior of the solution of the Cahn-Hilliard equation using "energy-type methods". We assume that the initial data has a "transition layer structure", i.e., u w ±1 except near finitely many transition points. We show that, in the limit as e —> 0, the solution maintains its transition layer structure, and the transition layers move slower than any power of e. §