Anomalous scaling for three‐dimensional Cahn‐Hilliard fronts

We prove the stability of the one‐dimensional kink solution of the Cahn‐Hilliard equation under d‐dimensional perturbations for d ≥ 3. We also establish a novel scaling behavior of the large‐time asymptotics of the solution. The leading asymptotics of the solution is characterized by a length scale proportional to t^1/3 instead of the usual t^1/2 scaling typical to parabolic problems. © 2004 Wiley Periodicals, Inc.     

Stability and convergence of the spectral Galerkin method for the Cahn‐Hilliard equation

A spectral Galerkin method in the spatial discretization is analyzed to solve the Cahn‐Hilliard equation. Existence, uniqueness, and stabilities for both the exact solution and the approximate solution are given. Using the theory and technique of a priori estimate for the partial differential equation, we obtained the convergence of the spectral Galerkin method and the error estimate between the approximate solution u_N(t) and the exact solution u(t). © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Finite difference approximate solutions for the Cahn‐Hilliard equation

In this article, we analyze a Crank‐Nicolson‐type finite difference scheme for the nonlinear evolutionary Cahn‐Hilliard equation. We prove existence, uniqueness and convergence of the difference solution. An iterative algorithm for the difference scheme is given and its convergence is proved. A linearized difference scheme is presented, which is also second‐order convergent. Finally a new difference method possess a Lyapunov function is presented. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 437–455, 2007     

Exponential attractors for a singularly perturbed Cahn‐Hilliard system

Our aim in this article is to give a construction of exponential attractors that are continuous under perturbations of the underlying semigroup. We note that the continuity is obtained without time shifts as it was the case in previous studies. Moreover, we obtain an explicit estimate for the symmetric distance between the perturbed and unperturbed exponential attractors in terms of the perturbation parameter. As an application, we prove the continuity of exponential attractors for a viscous Cahn‐Hilliard system to an exponential attractor for the limit Cahn‐Hilliard system. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic behavior of higher‐order Navier‐Stokes‐Cahn‐Hilliard systems

Our aim in this paper is to study the asymptotic behavior, in terms of finite‐dimensional attractors, for higher‐order Navier‐Stokes‐Cahn‐Hilliard systems. Such equations describe the evolution of a mixture of 2 immiscible incompressible fluids. We also give several numerical simulations.     

3‐D viscous Cahn–Hilliard equation with memory

We deal with the memory relaxation of the viscous Cahn–Hilliard equation in 3‐D, covering the well‐known hyperbolic version of the model. We study the long‐term dynamic of the system in dependence of the scaling parameter of the memory kernel ε and of the viscosity coefficient δ. In particular we construct a family of exponential attractors, which is robust as both ε and δ go to zero, provided that ε is linearly controlled by δ. Copyright © 2008 John Wiley & Sons, Ltd.     

Sixth‐order Cahn–Hilliard systems with dynamic boundary conditions

Our aim in this paper is to define dynamic boundary conditions for several sixth‐order Cahn–Hilliard systems. We then study the well‐posedness and the dissipativity of the systems derived. Copyright © 2014 John Wiley & Sons, Ltd.     

Robust exponential attractors for Cahn‐Hilliard type equations with singular potentials

Our aim in this article is to study the long time behaviour of a family of singularly perturbed Cahn‐Hilliard equations with singular (and, in particular, logarithmic) potentials. In particular, we are able to construct a continuous family of exponential attractors (as the perturbation parameter goes to 0). Furthermore, using these exponential attractors, we are able to prove the existence of the finite dimensional global attractor which attracts the bounded sets of initial data for all the possible values of the spatial average of the order parameter, hence improving previous results which required strong restrictions on the size of the spatial domain and to work on spaces on which the average of the order parameter is prescribed. Finally, we are able, in one and two space dimensions, to separate the solutions from the singular values of the potential, which allows us to reduce the problem to one with a regular potential. Unfortunately, for the unperturbed problem in three space dimensions, we need additional assumptions on the potential, which prevents us from proving such a result for logarithmic potentials. Copyright © 2004 John Wiley & Sons, Ltd.     

Slow dynamics for the hyperbolic Cahn‐Hilliard equation in one‐space dimension

The aim of this paper is to study the metastable properties of the solutions to a hyperbolic relaxation of the classic Cahn‐Hilliard equation in one‐space dimension, subject to either Neumann or Dirichlet boundary conditions. To perform this goal, we make use of an “energy approach," already proposed for various evolution PDEs, including the Allen‐Cahn and the Cahn‐Hilliard equations. In particular, we shall prove that certain solutions maintain a N‐transition layer structure for a very long time, thus proving their metastable dynamics. More precisely, we will show that, for an exponentially long time, such solutions are very close to piecewise constant functions assuming only the minimal points of the potential, with a finitely number of transition layers, which move with an exponentially small velocity.     

Infinite‐energy solutions for the Cahn–Hilliard equation in cylindrical domains

We give a detailed study of the infinite‐energy solutions of the Cahn–Hilliard equation in the 3D cylindrical domains in uniformly local phase space. In particular, we establish the well‐posedness and dissipativity for the case of regular potentials of arbitrary polynomial growth as well as for the case of sufficiently strong singular potentials. For these cases, we prove the further regularity of solutions and the existence of a global attractor. For the cases where we have failed to prove the uniqueness (e.g., for the logarithmic potentials), we establish the existence of the trajectory attractor and study its properties. Copyright © 2013 John Wiley & Sons, Ltd.     

Classical solvability of 1‐D Cahn–Hilliard equation coupled with elasticity

In this paper, we prove the classical solvability of a nonlinear 1‐D system of hyperbolic–parabolic type arising as a model of phase separation in deformable binary alloys. The system is governed by the nonstationary elasticity equation coupled with the Cahn–Hilliard equation. The existence proof is based on the application of the Leray–Schauder fixed point theorem and standard energy methods. Copyright © 2006 John Wiley & Sons, Ltd.     

A Cahn–Hilliard type equation with periodic gradient‐dependent potentials and sources

This paper is concerned with the existence, uniqueness and attractability of time periodic solutions of a Cahn–Hilliard type equation with periodic gradient‐dependent potentials and sources. Copyright © 2009 John Wiley & Sons, Ltd.     

Goal‐oriented error estimation for Cahn–Hilliard models of binary phase transition

A posteriori estimates of errors in quantities of interest are developed for the nonlinear system of evolution equations embodied in the Cahn–Hilliard model of binary phase transition. These involve the analysis of wellposedness of dual backward‐in‐time problems and the calculation of residuals. Mixed finite element approximations are developed and used to deliver numerical solutions of representative problems in one‐ and two‐dimensional domains. Estimated errors are shown to be quite accurate in these numerical examples. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Periodic solutions to the Cahn–Hilliard equation with constraint

This paper is concerned with the multidimensional Cahn–Hilliard equation with a constraint. The existence of periodic solutions of the problem is mainly proved under consideration by the viscosity approach. More precisely, with the help of the subdifferential operator theory and Schauder fixed point theorem, the existence of solutions to the approximation of the original problem is shown, and then the solution is obtained by using a passage‐to‐limit procedure based on a prior estimate. Copyright © 2015 John Wiley & Sons, Ltd.     

Global regular solutions to Cahn–Hilliard system coupled with viscoelasticity

In this paper we prove the existence and uniqueness of a global in time, regular solution to the Cahn–Hilliard system coupled with viscoelasticity. The system arises as a model, regularized by a viscous damping, of phase separation process in a binary deformable alloy quenched below a critical temperature. The key tools in the analysis are estimates of absorbing type with the property of exponentially time‐decreasing contribution of the initial data. Such estimates allow not only to prolong the solution step by step on the infinite time interval but also to conclude the existence of an absorbing set. Copyright © 2009 John Wiley & Sons, Ltd.     

A numerical scheme for the solution of viscous Cahn–Hilliard equation

In this paper, we present a numerical scheme for the solution of viscous Cahn–Hilliard equation. The scheme is based on Adomian's decomposition approach and the solutions are calculated in the form of a convergent series with easily computable components. Some numerical examples are presented. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Analysis of the operator splitting scheme for the Cahn‐Hilliard equation with a viscosity term

In this paper, we consider a second‐order fast explicit operator splitting method for the viscous Cahn‐Hilliard equation, which includes a viscosity term αΔu_t (α ∈ (0, 1)) described the influences of internal micro‐forces. The choice α = 0 corresponds to the classical Cahn‐Hilliard equation whilst the choice α = 1 recovers the nonlocal Allen‐Cahn equation. The fundamental idea of our method is to split the original problem into linear and nonlinear parts. The linear subproblem is numerically solved using a pseudo‐spectral method, and thus an ordinary differential equation is obtained. The nonlinear one is solved via TVD‐RK method. The stability and convergence are discussed in L²‐norm. Numerical experiments are performed to validate the accuracy and efficiency of the proposed method. Besides, a detailed comparison is made for the dynamics and the coarsening process of the metastable pattern for various values of α. Moreover, energy degradation and mass conservation are also verified.     

Convergence analysis for second‐order accurate schemes for the periodic nonlocal Allen‐Cahn and Cahn‐Hilliard equations

In this paper, we provide a detailed convergence analysis for fully discrete second‐order (in both time and space) numerical schemes for nonlocal Allen‐Cahn and nonlocal Cahn‐Hilliard equations. The unconditional unique solvability and energy stability ensures ? ⁴ stability. The convergence analysis for the nonlocal Allen‐Cahn equation follows the standard procedure of consistency and stability estimate for the numerical error function. For the nonlocal Cahn‐Hilliard equation, because of the complicated form of the nonlinear term, a careful expansion of its discrete gradient is undertaken, and an H ^?1 inner‐product estimate of this nonlinear numerical error is derived to establish convergence. In addition, an a priori bound of the numerical solution at the discrete level is needed in the error estimate. Such a bound can be obtained by performing a higher order consistency analysis by using asymptotic expansions for the numerical solution. Following the technique originally proposed by Strang (eg, 1964), instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an O (s ³+h ⁴) convergence in norm, in which s and h denote the time step and spatial mesh sizes, respectively. This in turn leads to the necessary bound under a standard constraint s ≤C h . Here, we also prove convergence of the scheme in the maximum norm under the same constraint.     

A Cahn–Hilliard model in bounded domains with permeable walls

In this article, we propose a model of phase separation in a binary mixture confined to a bounded region which may be contained within porous walls. The boundary conditions are derived from a mass conservation law and variational methods. Employing classical methods, that is, fixed point theorems and standard energy methods, we obtain the existence and uniqueness of a global solution to our problem. We then also compare our model of phase separation with other previous Cahn–Hilliard equations with homogeneous Neumann and dynamic boundary conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

Stabilized semi‐implicit spectral deferred correction methods for Allen–Cahn and Cahn–Hilliard equations

Stabilized semi‐implicit spectral deferred correction methods are constructed for the time discretization of Allen–Cahn and Cahn–Hilliard equations. These methods are unconditionally stable, lead to simple linear system to solve at each iteration, and can achieve high‐order time accuracy with a few iterations in each time step. Ample numerical results are presented to demonstrate the effectiveness of the stabilized semi‐implicit spectral deferred correction methods for solving the Allen–Cahn and Cahn–Hilliard equations. Copyright © 2013 John Wiley & Sons, Ltd.