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In this paper, we propose integrating factor (IF)‐based methods for solving the Cahn–Hilliard (CH) equation, which was originally proposed to describe the phase separation phenomenon for binary melted alloy below the critical temperature, and since then, it has appeared in many fields. Being a nonlinear equation, it is of great importance to develop efficient numerical methods for investigating the solution behavior of the CH equation. We present a few important linear and nonlinear schemes for the CH equation and obtain error estimates for those schemes. The integrators that we present here are of first and second order in time. We show numerical results to illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we consider an explicit fully‐discrete approximation of the Cahn–Hilliard–Cook (CHC) equation with additive noise, performed by a standard finite element method in space and a kind of nonlinearity‐tamed Euler scheme in time. The main result in this paper establishes strong convergence rates of the proposed scheme. The key ingredient in the proof of our main result is to employ uniform moment bounds for the numerical approximations. To the best of our knowledge, the main contribution of this work is the first result in the literature which establishes strong convergence for an explicit fully‐discrete finite element approximation of the CHC equation. Finally, numerical results are finally reported to confirm the previous theoretical findings. [ABSTRACT FROM AUTHOR]

The aim of this paper is to analyze a viscoelastic phase separation model. We derive a suitable notion of the relative energy taking into account the nonconvex nature of the energy law for the viscoelastic phase separation. This allows us to prove the weak–strong uniqueness principle. We will provide the estimates for the full model in two space dimensions. For a reduced model, we present the estimates in three space dimensions and derive conditional relative energy estimates. [ABSTRACT FROM AUTHOR]

In this article, we devote to the finite dimensional reduction for the hyperviscous Navier–Stokes equation with L2$$ {L}^2 $$ force and prove the existence of an N$$ N $$‐dimensional inertial manifold for this problem in 핋3. [ABSTRACT FROM AUTHOR]

It is well known that the classic Allen–Cahn equation satisfies the maximum bound principle (MBP), that is, the absolute value of its solution is uniformly bounded for all time by certain constant under suitable initial and boundary conditions. In this paper, we consider numerical solutions of the modified Allen–Cahn equation with a Lagrange multiplier of nonlocal and local effects, which not only shares the same MBP as the original Allen–Cahn equation but also conserves the mass exactly. We reformulate the model equation with a linear stabilizing technique, then construct first‐ and second‐order exponential time differencing schemes for its time integration. We prove the unconditional MBP preservation and mass conservation of the proposed schemes in the time discrete sense and derive their error estimates under some regularity assumptions. Various numerical experiments in two and three dimensions are also conducted to verify the theoretical results. [ABSTRACT FROM AUTHOR]

We consider the process of phase separation of a binary system under the influence of mechanical stress modelled by the Cahn–Larché system, where the mechanical deformation takes place on a macrosopic scale whereas the phase separation happens on a microscopic level. After linearisation, we prove existence and uniqueness of a weak solution by a Galerkin approach. As discretisation in space leads to a linear differential–algebraic system of equations, we adjust known solution theory for such equations to a weak setting. This approach may be of interest more generally for coupled elliptic–parabolic systems. A‐priori estimates enable us to pass to the homogenisation limit of the linear system rigorously using the concept of two‐scale convergence. A comparison with the formally homogenised full nonlinear problem shows that both systems lead to models of distributed‐microstructure type in the limit and that homogenisation and linearisation commutes. We consider the process of phase separation of a binary system under the influence of mechanical stress modelled by the Cahn–Larché system, where the mechanical deformation takes place on a macrosopic scale whereas the phase separation happens on a microscopic level. After linearisation, we prove existence and uniqueness of a weak solution by a Galerkin approach. As discretisation in space leads to a linear differential–algebraic system of equations, we adjust known solution theory for such equations to a weak setting.... [ABSTRACT FROM AUTHOR]

We analyze a diffuse interface model that couples a viscous Cahn–Hilliard equation for the phase function φ with a diffusion‐reaction equation for the nutrient concentration σ. The system under consideration also takes into account some important mechanisms like chemotaxis, active transport, and nonlocal interaction of Oono's type. When the spatial dimension is three, we prove the existence and uniqueness of a global weak solution to the model with singular potentials including the physically relevant logarithmic potential. Then we obtain some regularity properties of the weak solution when t > 0. In particular, with the aid of the viscous term ϵ∂tφ, we prove the so‐called instantaneous separation property of the phase variable φ such that it stays away from the pure states ±1 as long as t > 0. Next, we study longtime behavior of the system, by proving the existence of the global attractor in a suitable phase space and characterizing the ω‐limit set. Moreover, we show the existence of an exponential attractor, which implies that the global attractor is of finite fractal dimension. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Galerkin finite element method for the Cahn‐Hilliard‐Cook equation involving the Caputo‐type fractional derivative, which can be used to describe the interface phenomena for modeling the phase transitions with the random effects and the properties of shape memory. The regularity properties of mild solution to the given problem are presented, and a result concerning the convergence error estimate of the corresponding semidiscrete scheme is established. Finally, we construct the fully discrete scheme based on the approximations of the Mittag‐Leffler function, and the strong convergence error estimate of the proposed scheme is also studied. [ABSTRACT FROM AUTHOR]

Comparing with the classical local gradient flow and phase field models, the nonlocal models such as nonlocal Cahn–Hilliard equations equipped with nonlocal diffusion operator can describe more practical phenomena for modeling phase transitions. In this paper, we construct an accurate and efficient scalar auxiliary variable approach for the nonlocal Cahn–Hilliard equation with general nonlinear potential. The first contribution is that we have proved the unconditional energy stability for nonlocal Cahn–Hilliard model and its semi‐discrete schemes carefully and rigorously. Second, what we need to focus on is that the nonlocality of the nonlocal diffusion term will lead the stiffness matrix to be almost full matrix which generates huge computational work and memory requirement. For spatial discretizaion by finite difference method, we find that the discretizaition for nonlocal operator will lead to a block‐Toeplitz–Toeplitz‐block matrix by applying four transformation operators. Based on this special structure, we present a fast procedure to reduce the computational work and memory requirement. Finally, several numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes. [ABSTRACT FROM AUTHOR]

This paper is concerned with the compressible Cahn‐Hilliard‐Navier‐Stokes system. We establish a sufficient regularity condition such that every weak solution conserves its energy equality for every t > 0. Our approach is based on the commutator and mollification approximation. [ABSTRACT FROM AUTHOR]

This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial‐boundary problem for the nonlinear diffusion equation in an unbounded domain Ω ⊂ R N( N ∈ N), written as ∂ u ∂ t + ( − Δ + 1 ) β ( u ) = g in Ω × ( 0 , T ) , which represents the porous media, the fast diffusion equations, etc, where β is a single‐valued maximal monotone function on R, and T>0. In Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) existence and uniqueness of solutions for were directly proved under a growth condition for β even though the Stefan problem was excluded from examples of . This paper completely removes the growth condition for β by confirming Cauchy's criterion for solutions of the following approximate problem ε with approximate parameter ε>0: ∂ u ε ∂ t + ( − Δ + 1 ) ( ε ( − Δ + 1 ) u ε + β ( u ε ) + π ε ( u ε ) ) = g in Ω × ( 0 , T ) , which is called the Cahn‐Hilliard system, even if Ω ⊂ R N( N ∈ N) is an unbounded domain. Moreover, it can be seen that the Stefan problem excluded from Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) is covered in the framework of this paper. [ABSTRACT FROM AUTHOR]

We consider a Cahn-Hilliard-type equation with degenerate mobility and single-well potential of Lennard-Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. We give existence results for different classes of weak solutions. Moreover, we formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution for any spatial dimension together with the convergence to the weak solution for spatial dimension d=1. We present simulation results in 1 and 2 space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics. In this case, we find similar results to the ones obtained in standard phase ordering dynamics and we highlight the fact that the asymptotic behavior of the solution is dominated by the mechanism of growth by bulk diffusion. [ABSTRACT FROM AUTHOR]

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 ℓ4 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 [ABSTRACT FROM AUTHOR]

Abstract: In the present paper we derive a reduced order model utilizing proper orthogonal decomposition (POD‐ROM), for which we utilize adaptively obtained spatial snapshots. In a fully discrete setting this contains the challenge that the snapshots are vectors of different lengths. In order to handle this issue, we interprete the snapshots as elements of a common Hilbert space and consider the POD method from an infinite‐dimensional perspective. Thus, the inner product of pairs of snapshots can be computed explicitely, which enables us to build the reduced order model. This approach is applied to a phase field model which is described by a Cahn‐Hilliard equation. In the numerical examples we illustrate our appoach and compare a nonsmooth with a smooth free energy concerning the influence on the quality of the solution to the ROM. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

In this article, we deal with a rigorous error analysis for the finite element solutions of the two-dimensional Cahn-Hilliard equation with infinite time. The [ABSTRACT FROM AUTHOR]

The thermally induced phase separation can be used to prepare materials of various microstructured morphologies that define their applications. In this work, foams formed from a polystyrene-cyclohexanol solution are prepared, characterized, and modeled. To obtain a reliable thermodynamic description, cloud points are determined by an in-house fabricated thermooptical device. The Flory-Huggins lattice model combined with Hansen solubility theory is employed as input to the Cahn-Hilliard model of phase separation dynamics. The previously reported model is extended by incorporating polar interactions. The resulting experimental and predicted morphologies are compared and good prediction capabilities of the model regarding the resulting morphology and its characteristics are obtained. The experimental and modeling work extends the knowledge of the mechanism and the kinetics of foam morphology evolution for thermal and sound insulation applications. [ABSTRACT FROM AUTHOR]

We propose a practical estimation of a splitting parameter for a spectral method for the ternary Cahn-Hilliard system with a logarithmic free energy. We use Eyre's convex splitting scheme for the time discretization and a Fourier spectral method for the space variables. Given an absolute temperature, we find composition values that make the total free energy be minimum. Then, we find the splitting parameter value that makes the two split homogeneous free energies be convex on the neighborhood of the local minimum concentrations. For general use, we also propose a sixth-order polynomial approximation of the minimum concentration and derive a useful formula for the practical estimation of the splitting parameter in terms of the absolute temperature. The numerical tests are phase separation and total energy decrease with different temperature values. The linear stability analysis shows a good agreement between the exact and numerical solutions with an optimal value s. Various computational experiments confirm that the proposed splitting parameter estimation gives stable numerical results. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this note we recall two approaches to evolutionary Γ-convergence for classical gradient systems: evolutionary variational inequalities versus energy-dissipation principles. Following [1] we apply both approaches to homogenize Cahn-Hilliard-type equations with rapidly oscillating coefficients and quite general potential. Here the focus is on the Γ-convergence of the energy functional, which we prove without assuming any growth conditions for the convex part of the potential. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

The article discusses a study on an effective method to suppress the branching of self-organized molecular monolayer stripes obtained with the Langmuir–Blodgett (LB) technique by using prestructured surfaces It also discusses the use of Cahn–Hilliard model, and ways to control orientation of LB patterns.

A finite strain phase field fracture model is presented which treats phase segregation, fracture and the related electrochemical reaction in rechargeable Li-ion battery electrodes in a systematic manner. In particular, a regularized electrochemical reaction model is proposed to account for the reaction on the freshly generated diffusive fracture surface. It is shown that this model is equivalent to the reaction model on a sharp surface. With this regularized formulation, the reaction induced source term is added into the Cahn-Hilliard type evolution equation. Moreover, to ensure that hydrostatic pressure makes no contribution to fracture, a decomposition of the elastic strain energy is proposed. The fracture model is implemented with isogeometric finite element method and used to numerically study the interaction among phase interface, fracture and electrochemical reaction. Simulations show that phase interface can induce crack branching. The electrochemical reaction induced crack formation from micro pores is also observed. Detailed analysis demonstrates that the model takes reasonably into account phase segregation, fracture, electrochemical reaction, and especially their interplay. (© 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Our aim in this paper is to study, in term of finite dimensional exponential attractors, the Willmore regularization, (depending on a small regularization parameter β > 0), of two phase-field equations, namely, the Allen-Cahn and the Cahn-Hilliard equations. In both cases, we construct robust families of exponential attractors, that is, attractors that are continuous with respect to the perturbation parameter. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we continue the study of the hyperbolic relaxation of the Cahn-Hilliard-Oono equation with the sub-quintic non-linearity in the whole space [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For a large class of initial data, we prove the existence of classical solutions locally in time to a modified Hele-Shaw problem that takes elastic effects into account. The system arises as sharp interface model of a Cahn-Hilliard system coupled with linearized elasticity. By using the Hanzawa transformation, we can reduce the system to a single evolution equation for the height function. Then short time existence is proven by inverting the linearized operator and applying the contraction mapping principle. [ABSTRACT FROM AUTHOR]

We present an energy-stable, decoupled discrete scheme for a recent model (see [1]) supposed to describe electrokinetic phenomena in two-phase flow with general mass densities. This model couples momentum and Cahn-Hilliard type phase-field equations with Nernst-Planck equations for ion density evolution and an elliptic transmission problem for the electrostatic potential. The transport velocities in our scheme are based on the old velocity field updated via a discrete time integration of the force densities. This allows to split the equations into three blocks which can be treated sequentially: The phase-field equation, the equations for ion transport and electrostatic potential, and the Navier-Stokes type equations. By establishing a discrete counterpart of the continuous energy estimate, we are able to prove the stability of the scheme. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

This paper is concerned with the time optimal control problem governed by the internal controlled Cahn-Hilliard equation. We prove the existence of optimal controls. Moreover, we give necessary optimality conditions for an optimal control of our original problem by using the one of the approximate problems. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Crack propagation in brittle materials with anisotropic surface energy is important in applications involving single crystals, extruded polymers, or geological and organic materials. Furthermore, when this anisotropy is strong, the phenomenology of crack propagation becomes very rich, with forbidden crack propagation directions or complex sawtooth crack patterns. This problem interrogates fundamental issues in fracture mechanics, including the principles behind the selection of crack direction. Here, we propose a variational phase-field model for strongly anisotropic fracture, which resorts to the extended Cahn-Hilliard framework proposed in the context of crystal growth. Previous phase-field models for anisotropic fracture were formulated in a framework only allowing for weak anisotropy. We implement numerically our higher-order phase-field model with smooth local maximum entropy approximants in a direct Galerkin method. The numerical results exhibit all the features of strongly anisotropic fracture and reproduce strikingly well recent experimental observations. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Two dimensional diffuse interface model for a chemically reacting incompressible binary fluid in a bounded domain is considered. The corresponding evolution system consists of the Navier-Stokes equations for the (averaged) fluid velocity that are nonlinearly coupled with a convective Cahn-Hilliard-Oono type equation for the difference ψ of two fluid concentrations. The effects of a (reversible) chemical reaction is represented in the latter equation by an additional term of the form ε( ψ − c0), ε > 0. Here, c0 is the stationary spatial average of ψ, provided that, for example, no-slip and no-flux boundary conditions are considered. The mass is not necessarily conserved unless the spatial average of the initial datum for ψ coincides with c0. When ε = 0 (i.e., no chemical reaction), the model reduces to the well-known Cahn-Hilliard-Navier-Stokes system, which has been investigated by several authors. Here, we want to show that the global dynamic behavior of the system is robust with respect to ε. More precisely, we construct a family of exponential attractors, which is continuous with respect to ε. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY We present unconditionally energy-stable second-order time-accurate schemes for diffuse-interface (phase-field) models; in particular, we consider the Cahn-Hilliard equation and a diffuse-interface tumor-growth system consisting of a reactive Cahn-Hilliard equation and a reaction-diffusion equation. The schemes are of the Crank-Nicolson type with a new convex-concave splitting of the free energy and an artificial-diffusivity stabilization. The case of nonconstant mobility is treated using extrapolation. For the tumor-growth system, a semi-implicit treatment of the reactive terms and additional stabilization are discussed. For suitable free energies, all schemes are linear. We present numerical examples that verify the second-order accuracy, unconditional energy-stability, and superiority compared with their first-order accurate variants. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We prove regularity results and, as a consequence, uniqueness for a system of partial differential equations arising in the study of dynamic electrowetting phenomena and more general electrokinetic processes in three space dimensions. The system consists of Stokes equations coupled with equations for the motion of electric charges, Poisson equation for computing the electric field generated by such charges and a Cahn-Hilliard equation for a phase field describing two fluids with different material parameters. The deduction and existence of weak solutions for this system was established in an earlier paper (Christof Eck et al., On a phase-field model for electrowetting, Interfaces Free Bound. 11 (2), 259-290, (2009)). [ABSTRACT FROM AUTHOR]

In this work the Cahn-Hilliard-type diffusion in a hyperelastic solid will be studied and simulated using isogeometric analysis. The Cahn-Hilliard model is employed to account for the phase segregation phenomenon, in which a fourth-order partial differential operator exists. In order to deal with this high-order operator, isogeometric analysis is applied. Finally, simulation results of a disc is given. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

The lower bainitic transformation is highly dependent on carbon diffusion. Bainite consists of bainitic ferrite, residual austenite and carbides. The numerical modeling of the interaction between these phases and the carbon is extremely demanding. The goal of this work is to describe the formation of carbides in lower bainite. To model the evolution of a bainitic sheaf a phase-field model is coupled with a Cahn-Hilliard equation simulating the diffusion. The system of equations is solved using the finite element method. Numerical examples show the growth of the ferrite and the following uphill diffusion within this phase. At accumulation points of carbon, carbides are precipitated. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

The objective of this contribution is to study various mixed finite element schemes applied to classical and non-classical diffusion problems. The Fickean diffusion is considered using a three-field formulation where the concentration gradient and the diffusive flux are additionally chosen as independent variables. Furthermore, the non-classical diffusion model given by the Cahn-Hilliard equation is discussed where mixed finite elements are used to cast the fourth-order PDE into two second-order PDEs. A penalized version of a mixed formulation for the underlying second gradient model is given by specific micromorphic models. Another approach is a classical second order splitting method. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]